Independence polynomials of graphs

INDEPENDENCE POL YNOMIALS OF GRAPHS T AKA YUKI HIBI, SEL VI KARA, AND D ALENA VIEN Abstract. In this pap er, we study the indep endence polynomial P G ( x ) of a finite simple graph G , with emphasis on the ev aluation at x = − 1, symmetry , and its connection with the h -p olynomial of the edge ideal of G . F or big star graphs, w e determine exactly when P G ( − 1) is 0, 1, or − 1, characterize the pseudo-Gorenstein ∗ mem b ers, and show that there is a unique big star with symmetric indep endence p olynomial. W e also study graphs obtained from a graph H b y attac hing lea v es to selected v ertices. W e deriv e an explicit form ula for the resulting independence polynomial, determine the corresponding v alue at − 1, and pro ve that if ev ery v ertex of H receiv es at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly tw o lea ves. As an application, we obtain exact criteria for the v alues of P G ( − 1) and for the pseudo-Gorenstein ∗ mem b ers of caterpillar graphs. F or co chordal graphs, w e classify all symmetric indep endence p olynomials. Finally , for connected graphs on n v ertices with small indep endence n umbers, we determine the exact range of p ossible v alues of P G ( − 1). Intr oduction F or a finite simple graph G , the indep endenc e p olynomial of G is P G ( x ) = α ( G ) X i =0 g i ( G ) x i , where g i ( G ) denotes the num b er of indep endent sets of G of size i . This p olynomial is one of the basic enumerativ e in v arian ts attached to a graph. Bey ond its purely com binatorial meaning, the indep endence p olynomial appears as the partition function of the hard-core mo del in statistical physics [ 7 ] and its zeros play an imp ortan t role in questions related to the Lov´ asz lo cal lemma and zero-free regions from probabilistic com binatorics [ 21 ]. Th us indep endence p olynomials sit at the intersection of graph enumeration, statistical ph ysics, probabilistic combinatorics, and comm utativ e algebra; see, for example, [ 2 , 8 , 13 , 16 ]. In this pap er, w e fo cus on tw o asp ects of P G ( x ): the ev aluation at x = − 1 and symmetry . The v alue P G ( − 1) is the alternating count of indep enden t sets, equiv alently the negative of the reduced Euler c haracteristic of the indep endence complex Ind( G ). This quantit y has b een studied from sev eral p oints of view, including b ounds in terms of graph structure, decycling n umber, and realizabilit y problems; see [ 3 , 4 , 17 , 19 ]. The second theme, symmetry of the indep endence p olynomial, b elongs to the broader study of palindromicit y , unimo dalit y , log- conca vity , and ro ots. There is a substan tial literature on constructions that force symmetry , notably the t wo-whisk er construction of Stev ano vi´ c and its later extensions and refinemen ts; 2020 Mathematics Subje ct Classific ation. 05C38,05C31. Key wor ds and phr ases. independence p olynomials of graphs, chordal graphs, co chordal graphs. The present pap er was completed while the first author stay ed at Bryn Mawr College, P ennsylv ania, F ebruary 28 – Marc h 21, 2026. 1 2 see [ 18 , 20 , 22 ]. Ev en for trees, these questions remain subtle: for example, if T is a tree, then P T ( − 1) ∈ {− 1 , 0 , 1 } by [ 17 ], y et it is far from clear whic h trees realize whic h v alue. A second source of motiv ation to study independence p olynomials comes from comm utative algebra. Let I ( G ) b e the edge ideal of G , and let h G ( t ) denote the h -p olynomial of S/I ( G ). It was discussed b y the authors in [ 13 ] that the v alue P G ( − 1) controls the top co efficien t of h G ( t ). Moreo ver, the m ultiplicit y of − 1 as a ro ot of P G ( x ), denoted b y M ( G ), con trols the degree of the h -p olynomial b y [ 2 , Theorem 4.4], deg h G ( t ) = α ( G ) − M ( G ) . Th us the multiplicit y of − 1 as a ro ot of P G ( x ) determines the a -inv arian t of S/I ( G ). Addi- tionally , the condition that G b e pseudo-Gorenstein ∗ , a new concept introduced and studied b y the authors in [ 13 ], is equiv alen t to the simple identit y P G ( − 1) = ( − 1) α ( G ) . This mak es the ev aluation at − 1 a natural meeting p oin t b et ween the combinatorics of indep enden t sets and the algebra of edge ideals. Our first fo cus is on big star gr aphs , obtained b y gluing several paths at a common center. F or these graphs, we sho w that the v anishing of P G ( − 1) is completely determined b y congruence classes mo dulo 3, while the sign in the nonzero cases is controlled by congruence classes mo dulo 6; see Theorems 2.4 and 2.5 . W e then obtain a complete description of pseudo- Gorenstein ∗ big stars in Corollary 2.8 . W e also study symmetry and prov e in Theorem 2.9 that, among all big stars, there is exactly one with symmetric indep endence p olynomial. Our second fo cus is on the effects of whiskering , attaching leav es. In Prop osition 3.1 w e pro vide a general form ula for the indep endence p olynomial of a graph obtained from a base graph H b y attaching f i lea ves to selected v ertices x i . When the p olynomial is ev aluated at − 1, a striking simplification o ccurs: only the set C ⊆ V ( H ) of v ertices that actually receiv e lea ves matters. More precisely , P G ( − 1) = 0 unless C is an indep endent set of H , and when C is indep enden t w e obtain P G ( − 1) = ( − 1) | C | P H − N H [ C ] ( − 1) . W e also prov e in Theorem 3.2 that if every v ertex of H receives at least one leaf, then the resulting graph has symmetric indep endence p olynomial if and only if exactly t wo lea ves are attac hed to each v ertex. This recov ers Stev anovi ´ c’s tw o-whisk er construction [ 22 ], which w as later sho wn to yield not only symmetry but also unimo dalit y [ 20 ], and sho ws that in our leaf-attac hment setting it is in fact the unique symmetric case. As an application of the whisk ering, w e study c aterpil lar gr aphs , obtained b y attaching leav es to paths. Enco ding the lo cations of the legs along the path leads to a pro duct formula for P G ( − 1) in Theorem 3.5 . F rom this we determine when 0 , ± 1 occurs; see Corollaries 3.6 and 3.9 . In addition, we characterize pseudo-Gorenstein ∗ caterpillars in Corollary 3.10 . W e also study c o chor dal gr aphs , equiv alently graphs whose complemen ts are c hordal. This class is natural b oth combinatorially and algebraically: indep enden t sets of G are cliques of G , and co chordal graphs play a cen tral role in the study of edge ideals with linear resolutions [ 6 , 11 ]. Since clique complexes of chordal graphs are quasi-forests [ 12 ], the indep endence p olynomial of a co chordal graph can b e approac hed through the b -sequence description of 3 quasi-forests from [ 10 ]. Using [ 10 ], we pro ve that a co chordal graph G with d = α ( G ) ≥ 2 has symmetric indep endence p olynomial if and only if P G ( x ) = (1 + x ) d + mx (1 + x ) d − 2 for some m ≥ 0. In particular, ev ery symmetric indep endence p olynomial of a co c hordal graph is automatically unimo dal. W e also recall that P G ( − 1) = 1 − k where k is the n um b er of connected comp onen ts of G ; see Corollary 4.5 . Finally , w e inv estigate how large or small P G ( − 1) can b e when the indep endence n um b er is small. F or connected graphs on n v ertices with α ( G ) ≤ 2, Theorem 5.4 sho ws that ev ery in teger in the in terv al  − ( n − 1) ,  ( n − 2) 2 4  − 1  o ccurs as P G ( − 1). The upp er b ound comes from Man tel’s theorem applied to the comple- men t, while the full interv al is realized b y an explicit connected c hordal family built from t w o cliques. In this wa y , our final section ma y b e viewed as a counterpart to earlier realizability results for P G ( − 1), such as the connected constructions of Cutler and Kahl from [ 3 ]. 1. Preliminaries Throughout the pap er, all graphs are finite and simple. F or a graph G , w e write V ( G ) and E ( G ) for its v ertex and edge sets, α ( G ) for its independence num b er, and G for its complemen t. W e denote b y Ind( G ) the indep endence complex of G , i.e. the simplicial complex whose faces are the indep enden t subsets of V ( G ). F or W ⊆ V ( G ), let N G ( W ) and N G [ W ] = W ∪ N G ( W ) denote the op en and closed neigh b orho o ds of W , and let G − W b e the induced subgraph on V ( G ) \ W . In particular, G − v := G − { v } and G − N [ v ] := G − N G [ v ]. W e write G ⊔ H for the disjoint union of graphs G and H . F or n ≥ 0, let P n denote the path on n v ertices, with the con ven tion that P 0 is the empt y graph. The indep endence p olynomial of G is P G ( x ) = α ( G ) X i =0 g i ( G ) x i = X F ∈ Ind( G ) x | F | , where g i ( G ) is the n um b er of indep endent sets of G of size i . When the graph is clear from con text, w e simply write g i . W e sa y that P G ( x ) is symmetric if x α ( G ) P G (1 /x ) = P G ( x ) , equiv alently , if g i ( G ) = g α ( G ) − i ( G ) for all i . W e repeatedly use the m ultiplicativity P G ⊔ H ( x ) = P G ( x ) P H ( x ) . Let G ha ve v ertex set { x 1 , . . . , x n } , let S = K [ x 1 , . . . , x n ], and let I ( G ) = ( x i x j : { x i , x j } ∈ E ( G )) b e the edge ideal of G . 4 Recall that h G ( t ) denotes the h -p olynomial of S /I ( G ) and h G ( t ) := h S/I ( G ) ( t ) = α ( G ) X i =0 h i ( G ) t i , where h i ( G ) = 0 for i > deg h G ( t ). W e denote the a -in v arian t of S/I ( G ) b y a ( G ) where a ( G ) := a ( S/I ( G )) = deg h G ( t ) − α ( G ) . Lastly , w e use M ( G ) to denote the multiplicit y of x = − 1 as a ro ot of the indep endence p olynomial P G ( x ). W e no w recall a collection of standard results and consequences (see [ 13 ]). Prop osition 1.1. L et α = α ( G ) . Then h G ( t ) = (1 − t ) α P G  t 1 − t  . In p articular, h α ( G ) = ( − 1) α P G ( − 1) . The next theorem, pro ved in [ 2 , Theorem 4.4], relates the degree of h G ( t ) to the m ultiplicit y of − 1 as a ro ot of P G ( x ). Theorem 1.2. [ 2 , The or em 4.4] F or every finite simple gr aph G , deg h G ( t ) = α ( G ) − M ( G ) . Th us, it follo ws from the ab ov e result that a ( G ) = − M ( G ) for every finite simple graph G . Moreo ver, a ( G ) = 0 if and only if P G ( − 1)  = 0. Next, we recall the definition of pseudo-Gorenstein ∗ graphs from [ 13 ]. Definition 1.3. Let G b e a finite simple graph and let α = α ( G ). W e say that G is pseudo-Gor enstein ∗ if a ( G ) = 0 and h α ( G ) = 1. It then follo ws that a graph G is pseudo-Gorenstein ∗ if and only if P G ( − 1) = ( − 1) α ( G ) . The follo wing standard v ertex-deletion recursion for indep endence polynomials is extremely useful for computations. Lemma 1.4. L et G b e a finite simple gr aph and let v ∈ V ( G ) . Then P G ( x ) = P G − v ( x ) + xP G − N [ v ] ( x ) . F or later reference, we record the follo wing general fact ab out the v alue of the indep endence p olynomial of a tree at x = − 1. Theorem 1.5. [ 17 , The or em 2.4] If T is a tr e e, then P T ( − 1) ∈ {− 1 , 0 , 1 } . The v alues of P P n ( − 1) for paths hav e been studied in the literature; see, for example, [ 14 ]. 5 Lemma 1.6. L et p n := P P n ( − 1) for n ≥ 0 . Then p 0 = 1 , p 1 = 0 , p n = p n − 1 − p n − 2 for n ≥ 2 . Conse quently, p n is p erio dic of p erio d 6 , and p n =        1 , n ≡ 0 , 5 (mo d 6) , 0 , n ≡ 1 , 4 (mo d 6) , − 1 , n ≡ 2 , 3 (mod 6) . 2. Big St ars In this section we study the independence p olynomial of big star graphs, obtained b y gluing sev eral paths at a common center. This family pro vides a natural testing ground in whic h the in teraction b et ween lo cal path structure and global branching can still b e analyzed explicitly . Using the v ertex-deletion recursion at the cen ter, together with the p erio dic behavior of P P n ( − 1), we determine exactly when the v alue of the indep endence p olynomial at − 1 is 0, 1, or − 1. W e then cha racterize pseudo-Gorenstein ∗ big stars. Finally , we turn to symmetry and show that there is only one big star whose indep endence polynomial is symmetric. Definition 2.1. Let q ≥ 3 b e an in teger and n 1 , . . . , n q b e p ositiv e in tegers. Let Γ i denote the path on the v ertex set V i = { x, x ( i ) 1 , . . . , x ( i ) n i } for 1 ≤ i ≤ q . Γ i is a path of length n i for eac h i . W e assume V i ∩ V j = { x } for i  = j . Let G ( n 1 , . . . , n q ) = Γ 1 ∪ · · · ∪ Γ q . Using the terminology of [ 1 , Definition 5.9] w e call G ( n 1 , . . . , n q ) a big star graph with a center vertex x . F or the rest of this section, let r := |{ i : n i is o dd }| and let x b e the cen ter vertex of G ( n 1 , . . . , n q ). Lemma 2.2. L et G ( n 1 , . . . , n q ) b e a big star gr aph. Then P G ( x ) = q Y i =1 P P n i ( x ) + x q Y i =1 P P n i − 1 ( x ) . Pr o of. W rite G := G ( n 1 , . . . , n q ) and let x b e its center v ertex. By deleting the cen ter vertex x , we obtain G − x = q G i =1 P n i , G − N [ x ] = q G i =1 P n i − 1 . Hence, by Lemma 1.4 and m ultiplicativit y on disjoint unions, P G ( x ) = q Y i =1 P P n i ( x ) + x q Y i =1 P P n i − 1 ( x ) with the con ven tion P P 0 ( x ) = 1. □ Remark 2.3. It follo ws from Lemma 2.2 that P G ( − 1) = q Y i =1 p n i − q Y i =1 p n i − 1 , 6 where p m := P P m ( − 1) as in Lemma 1.6 . F rom this w e need only the follo wing three conse- quences of Lemma 1.6 to determine when P G ( − 1) is either 0 , 1 or − 1: p m = 0 ⇐ ⇒ m ≡ 1 (mo d 3) , p m − 1 = 0 ⇐ ⇒ m ≡ 2 (mo d 3) , and m ≡ 0 (mod 3) = ⇒ p m = p m − 1 ∈ {± 1 } . W e start with the P G ( − 1) = 0 case. Theorem 2.4. L et G := G ( n 1 , . . . , n q ) b e a big star gr aph. Then P G ( − 1) = 0 if and only if either (1) n i ≡ 0 (mo d 3) for al l i , or (2) ther e exist i  = j such that n i ≡ 1 (mo d 3) and n j ≡ 2 (mo d 3) . Pr o of. F ollowing the discussion from Remark 2.3 , we ha ve P G ( − 1) = 0 if and only if q Y i =1 p n i = q Y i =1 p n i − 1 . Supp ose first that there exist i  = j suc h that n i ≡ 1 (mo d 3) and n j ≡ 2 (mo d 3). Then p n i = 0, so the first pro duct is zero, and p n j − 1 = 0, so the second pro duct is zero. Hence P G ( − 1) = 0. Next, supp ose that n i ≡ 0 (mo d 3) for all i . The n p n i = p n i − 1 for ev ery i . Therefore Q q i =1 p n i = Q q i =1 p n i − 1 . Hence again P G ( − 1) = 0. Con versely , assume that P G ( − 1) = 0. If b oth pro ducts are zero, then some factor p n i m ust b e zero, so n i ≡ 1 (mo d 3) for some i , and some factor p n j − 1 m ust b e zero, so n j ≡ 2 (mo d 3) for some j . Thus condition (ii) holds. If b oth pro ducts are nonzero, then no p n i is zero and no p n i − 1 is zero. Hence no n i ≡ 1 (mod 3) and no n i ≡ 2 (mod 3). Therefore n i ≡ 0 (mo d 3) for all i , so condition (i) holds. Th us P G ( − 1) = 0 if and only if either (i) or (ii) holds. As observed at the b eginning, this is equiv alent to a ( G ) < 0. □ It is known by Theorem 1.5 that P G ( − 1) ∈ {− 1 , 0 , 1 } . After identifying when P G ( − 1) = 0, our goal is to determine for whic h big stars we hav e P G ( − 1) = 1 and P G ( − 1) = − 1. As w e ha ve seen in Theorem 2.4 , mo d 3 data determines exactly when P G ( − 1) = 0 happ ens. Ho wev er, mo d 3 do es not determine its sign when P G ( − 1)  = 0. As we will see, the sign is con trolled b y congruence classes mo dulo 6 together with a parity condition. Theorem 2.5. L et G = G ( n 1 , . . . , n q ) b e a big star gr aph. L et c k := |{ i : n i ≡ k (mo d 6) }| for k ∈ { 0 , 1 , 2 , 3 , 4 , 5 } . Then P G ( − 1) =        ( − 1) c 2 + c 3 , if c 1 = c 4 = 0 and c 2 + c 5 > 0 , ( − 1) c 3 + c 4 +1 , if c 2 = c 5 = 0 and c 1 + c 4 > 0 , 0 , otherwise . 7 Pr o of. As in Remark 2.3 , w e ha ve P G ( − 1) = Q q i =1 p n i − Q q i =1 p n i − 1 where n i mo d 6 0 1 2 3 4 5 p n i 1 0 − 1 − 1 0 1 p n i − 1 1 1 0 − 1 − 1 0 No w consider the t wo pro ducts. If c 1 + c 4 > 0, then Q i p n i = 0. If c 2 + c 5 > 0, then Q i p n i − 1 = 0. So there are four cases to consider. Case 1: c 1 + c 4 > 0 and c 2 + c 5 > 0. Then b oth pro ducts are zero and P G ( − 1) = 0. Case 2: c 1 + c 4 = 0 and c 2 + c 5 = 0. Then ev ery n i ≡ 0 or 3 (mo d 6) and p n i = p n i − 1 ∈ {± 1 } for all i . So the t wo pro ducts are equal. Therefore P G ( − 1) = 0. Case 3: c 1 + c 4 = 0 and c 2 + c 5 > 0. Then Q i p n i − 1 = 0, while Q q i =1 p n i = ( − 1) c 2 + c 3 b ecause p n i = − 1 only when n i ≡ 2 or 3 (mo d 6). Hence P G ( − 1) = ( − 1) c 2 + c 3 . Case 4: c 2 + c 5 = 0 and c 1 + c 4 > 0. Then Q i p n i = 0, while Q q i =1 p n i − 1 = ( − 1) c 3 + c 4 b ecause p n i − 1 = − 1 only when n i ≡ 3 or 4 (mo d 6). Hence P G ( − 1) = − ( − 1) c 3 + c 4 = ( − 1) c 3 + c 4 +1 . □ The following characterizes when P G ( − 1) = 1 and when P G ( − 1) = − 1 . Corollary 2.6. L et G = G ( n 1 , . . . , n q ) b e a big star gr aph. (1) P G ( − 1) = 1 if and only if either (a) n i ≡ 1 (mo d 3) for al l i , at le ast one n i ≡ 2 (mo d 3) and c 2 + c 3 is even, or (b) n i ≡ 2 (mo d 3) for al l i , at le ast one n i ≡ 1 (mo d 3) and c 3 + c 4 is o dd. (2) P G ( − 1) = − 1 if and only if either (a) n i ≡ 1 (mo d 3) for al l i , at le ast one n i ≡ 2 (mo d 3) and c 2 + c 3 is o dd, or (b) n i ≡ 2 (mo d 3) for al l i , at le ast one n i ≡ 1 (mo d 3) and c 3 + c 4 is even. W e no w determine the independence n umber of big star graphs. Lemma 2.7. L et G ( n 1 , . . . , n q ) b e a big star gr aph. Then α  G ( n 1 , . . . , n q )  = q X i =1 ⌊ n i / 2 ⌋ + max { 1 , r } wher e r = |{ i : n i is o dd }| . 8 Pr o of. W rite G := G ( n 1 , . . . , n q ) with the center v ertex x . A maxim um indep enden t set of G either contains x , or do es not contain x . If it contains x , then on the i th arm we ma y c ho ose at most α ( P n i − 1 ) =  n i − 1 2  = j n i 2 k additional v ertices. Hence the largest suc h indep endent set has size 1 + P q i =1 ⌊ n i / 2 ⌋ . If it do es not con tain x , then on the i th arm we may choose at most α ( P n i ) = ⌈ n i / 2 ⌉ v ertices. Hence the largest such indep endent set has size P q i =1 ⌈ n i / 2 ⌉ . Therefore α ( G ) = max ( 1 + q X i =1 ⌊ n i / 2 ⌋ , q X i =1 ⌈ n i / 2 ⌉ ) = q X i =1 ⌊ n i / 2 ⌋ + max { 1 , r } where the last equality follows from P q i =1 ⌈ n i / 2 ⌉ = P q i =1 ⌊ n i / 2 ⌋ + r . □ W e are no w ready to describ e whic h big star graphs are pseudo-Gorenstein ∗ using Theo- rem 2.5 and Lemma 2.7 . Corollary 2.8. L et G = G ( n 1 , . . . , n q ) b e a big star gr aph. Then G is pseudo-Gor enstein ∗ if and only if one of the fol lowing holds: (1) n i ≡ 1 (mo d 3) for al l i , at le ast one n i ≡ 2 (mo d 3) , and α ( G ) ≡ c 2 + c 3 (mo d 2) ; (2) n i ≡ 2 (mo d 3) for al l i , at le ast one n i ≡ 1 (mo d 3) , and α ( G ) ≡ c 3 + c 4 + 1 (mo d 2) . Lastly , w e determine which big stars hav e symmetric indep endence p olynomials. Surpris- ingly , exactly one big star has this property . Theorem 2.9. L et G = G ( n 1 , . . . , n q ) b e a big star. Then P G ( x ) is symmetric if and only if G ∼ = G (1 , 1 , 5) . Pr o of. Let α := α ( G ). If P G ( x ) is symmetric, then g α = g 0 = 1. This means G has a unique maxim um indep enden t set. Recall that r = |{ i : n i is o dd }| . If r = 1, then there is a maxim um indep enden t set con taining the center and also one a voiding the cen ter, contradiction. If 2 ≤ r ≤ q − 1, then every maximum indep enden t set av oids the cen ter, and each even arm P 2 a con tributes at least tw o maxim um choices, again a contradiction. Hence either all arms are ev en or all arms are o dd. Supp ose first that n i = 2 a i for all i . In this case, g 1 = 1 + 2 P q i =1 a i and α = 1 + P q i =1 a i b y Lemma 2.7 . An indep enden t set of size α − 1 either av oids the center or contains it. If it av oids the cen ter, then on the i th arm x ( i ) 1 − x ( i ) 2 − · · · − x ( i ) 2 a i w e m ust c ho ose a maxim um indep enden t set of P 2 a i . Thus eac h arm P 2 a i con tribute exactly a i v ertices and the num b er of indep enden t sets of size a i in P 2 a i is  2 a i +1 − a i a i  = a i + 1. Th us these con tribute Q q i =1 ( a i + 1). If it con tains the cen ter, then x ( i ) 1 is forbidden on ev ery arm. So on the i th arm we may only c ho ose v ertices from x ( i ) 2 − x ( i ) 3 − · · · − x ( i ) 2 a i ∼ = P 2 a i − 1 . Since the cen ter already contributes one vertex, the arms together m ust contribute P q i =1 a i − 1. Hence exactly one arm, say the j th, contributes a j − 1 v ertices, while every other arm contributes a i , necessarily the unique 9 maxim um indep enden t set { x ( i ) 2 , x ( i ) 4 , . . . , x ( i ) 2 a i } . On the exceptional j th arm, the n umber of indep enden t sets of size a j − 1 in P 2 a j − 1 is  (2 a j − 1)+1 − ( a j − 1) a j − 1  =  a j +1 2  . Th us these contribute P q j =1  a j +1 2  . Therefore g α − 1 = q Y i =1 ( a i + 1) + q X i =1  a i + 1 2  . Since  a i +1 2  ≥ a i and, b ecause q ≥ 3 and eac h a i ≥ 1, w e ha ve Q q i =1 ( a i + 1) > 1 + P q i =1 a i . Th us, g α − 1 > 1 + 2 P q i =1 a i = g 1 , contradicting symmetry . This means all arms are o dd. Let n i = 2 a i + 1 for all i . By Lemma 2.7 , w e hav e α = q + P q i =1 a i . Now let I b e an indep enden t set containing the center x . Then x ( i ) 1 / ∈ I for ev ery i , so on the i th arm w e may c ho ose v ertices only from x ( i ) 2 − x ( i ) 3 − · · · − x ( i ) 2 a i +1 ∼ = P 2 a i . Hence at most a i v ertices from that arm. Therefore, since q ≥ 3, w e get | I | ≤ 1 + P q i =1 a i ≤ α − 2. Th us no independent set of size α − 1 con tains the cen ter. It follo ws that every indep endent set coun ted b y g α − 1 a voids the cen ter. On the i th arm, the maxim um possible con tribution is a i + 1. Hence such a set m ust b e maximal on all but exactly one arm, and on the remaining arm, say the i th, it must hav e size a i . F or j  = i , the unique maximum indep enden t set of P 2 a j +1 is { x ( j ) 1 , x ( j ) 3 , . . . , x ( j ) 2 a j +1 } , while the num b er of independent sets of size a i in P 2 a i +1 is  (2 a i +1)+1 − a i a i  =  a i +2 2  . Summing o ver the c hoice of the exceptional arm gives g α − 1 = q X i =1  a i + 2 2  . Since P G ( x ) is symmetric of degree α , we ha v e g α − 1 = q X i =1  a i + 2 2  = q + 1 + 2 q X i =1 a i = g 1 . Since  a i +2 2  =  a i 2  + 2 a i + 1, the equalit y b ecomes P q i =1  a i 2  = 1. This equality forces exactly one term to b e 1 and all the others to b e 0. Hence, after reordering, exactly one a i is 2, and ev ery other a j is 0 or 1. Since n i = 2 a i + 1, this means that exactly one arm has length 5, and every other arm has length 1 or 3. If q = 3, then the only p ossibilities are G (1 , 1 , 5), G (1 , 3 , 5), and G (3 , 3 , 5), and a direct computation shows that only G (1 , 1 , 5) is symmetric. No w assume q ≥ 4. F rom the previous paragraph, after reordering w e ma y write G ∼ = G (1 , . . . , 1 | {z } b , 3 , . . . , 3 | {z } t , 5) where q = b + t + 1 and α = b + 2 t + 3. Set S ( x ) := (1 + x ) b (1 + 3 x + x 2 ) t . Then, we hav e P G ( x ) = S ( x ) (1 + 5 x + 6 x 2 + x 3 ) | {z } A ( x ) + x (1 + 4 x + 3 x 2 )(1 + 2 x ) t | {z } B ( x ) . Since S ( x ) is symmetric of degree b + 2 t = α − 3, w e ha ve x α − 3 S ( x − 1 ) = S ( x ). Then [ x α − k ] S ( x ) A ( x ) = [ x k ] S ( x ) x 3 A ( x − 1 ) . 10 Therefore [ x k ] S ( x ) A ( x ) − [ x α − k ] S ( x ) A ( x ) = [ x k ] S ( x )  A ( x ) − x 3 A ( x − 1 )  where A ( x ) − x 3 A ( x − 1 ) = x ( x − 1). Hence [ x k ] S ( x ) A ( x ) − [ x α − k ] S ( x ) A ( x ) = [ x k ]  x ( x − 1) S ( x )  . Next, deg B = t + 3 = α − ( q − 1). So if k ≤ q − 2, then α − k > deg B . Th us [ x α − k ] B ( x ) = 0. It follows that for every k ≤ q − 2, w e ha ve g k − g α − k = [ x k ]  B ( x ) + x ( x − 1) S ( x )  . Consider k = 2. One can deduce the co efficien ts of x 2 in B ( x ) and x ( x − 1) S ( x ) to conclude that g 2 − g α − 2 = 6 − q . If P G ( x ) is symmetric, then q = 6 and b + t = 5. No w consider k = 3 and make use of b = 5 − t in deducing the co efficient of x 3 in B ( x ) and x ( x − 1) S ( x ). So, w e ha v e g 3 − g α − 3 = t − 2. Symmetry forces t = 2. Hence b = 3. Th us the only p ossibility with q ≥ 4 is G ∼ = G (1 , 1 , 1 , 3 , 3 , 5). A direct computation giv es the follo wing p olynomial, which is not symmetric: P G ( x ) = 1 + 15 x + 91 x 2 + 296 x 3 + 577 x 4 + 714 x 5 + 575 x 6 + 296 x 7 + 91 x 8 + 15 x 9 + x 10 . Therefore P G ( x ) is symmetric if and only if G ∼ = G (1 , 1 , 5). □ 3. Whiskering and Independence Pol ynomials In this section, w e study the effect of whiskering, that is, attac hing leav es to select vertices, on the independence p olynomial. W e deriv e a general form ula for the independence p olynomial, obtain a simple expression for its v alue at − 1, and sho w that if ev ery v ertex of the base graph receiv es at least one leaf, then symmetry o ccurs exactly when each vertex receives tw o lea v es. Prop osition 3.1. L et H b e a finite simple gr aph on vertic es x 1 , . . . , x n , and let f 1 , . . . , f n b e nonne gative inte gers. L et G b e the gr aph obtaine d fr om H by attaching f i le af vertic es to x i for e ach i ∈ [ n ] . S et C := { x i : f i > 0 } ⊆ V ( H ) . Then P G ( x ) = X S ⊆ V ( H ) S is independent in H x | S | Y x i / ∈ S (1 + x ) f i . Mor e over, we have the fol lowing statements: (a) If C is not an indep endent set of H , then P G ( − 1) = 0 . (b) If C is an indep endent set of H , then P G ( − 1) = ( − 1) | C | P H − N H [ C ] ( − 1) . (c) One has α ( G ) = P n i =1 f i + α ( H − C ) . Pr o of. Let L i denote the set of leav es attac hed to x i and | L i | = f i . Given an indep endent set I of G , let S := I ∩ V ( H ) = { x i ∈ V ( H ) : x i ∈ I } . Then S is an indep endent set of H . Con versely , if S is indep endent in H , then for eac h i ∈ S no v ertex of L i ma y b e chosen, while for each i / ∈ S an y subset of L i ma y b e chosen. Hence the total con tribution of all indep enden t sets I with this fixed set S is x | S | Y x i / ∈ S (1 + x ) f i , 11 whic h pro ves the display ed form ula for P G ( x ). No w fo cus on P G ( − 1). Notice that for an indep enden t set S of H , whenev er i ∈ C (i.e. f i > 0) but i / ∈ S , then the corresp onding summand for S is zero. So, for a summand to surviv e, ev ery x i ∈ C m ust b e con tained in S . So, a summand is nonzero if and only if f i = 0 for ev ery i / ∈ S , i.e. if and only if C ⊆ S . Th us, if C is not indep endent in H , then no indep endent set S can con tain C . Hence P G ( − 1) = 0. This pro ves (a). Assume no w that C is indep endent in H . Then the surviving sets S are exactly those of the form S = C ⊔ T where T is an independent set of H − N H [ C ]. Therefore P G ( − 1) = X T ∈ Ind( H − N H [ C ]) ( − 1) | C | + | T | = ( − 1) | C | P H − N H [ C ] ( − 1) , pro ving (b). Finally , let I b e a maximum indep endent set of G . If x i ∈ I for some i with f i > 0, then replacing x i b y all vertices of L i preserv es indep endence and do es not decrease cardinality . Rep eating this for each i , we obtain a maxim um indep endent set J con taining no v ertex of C . Since eac h leaf in L i is adjacent only to x i , maximality forces L i ⊆ J for every i with f i > 0. Th us J consists of all P i f i attac hed lea ves together with an indep endent set of H − C . Con versely , any indep endent set of H − C , together with all attac hed lea ves, is indep enden t in G . Hence α ( G ) = n X i =1 f i + α ( H − C ) , whic h pro ves (c). □ In what follows, w e discuss when the whiskering op eration yields a symmetric indep endence p olynomial. This result generalizes and reco vers one of the constructions of Stev anovi ´ c’s from [ 22 ] guaran teeing symmetric indep endence p olynomials. Theorem 3.2. Assume in the setting of Pr op osition 3.1 that f i ≥ 1 for every i ∈ [ n ] , so C = V ( H ) . Then P G ( x ) is symmetric if and only if f 1 = · · · = f n = 2 . Pr o of. Set F := P n i =1 f i . Since C = V ( H ), Prop osition 3.1 (c) yields α ( G ) = F . Again by Prop osition 3.1 , (1) P G ( x ) = X S ∈ Ind( H ) C S ( x ) where C S ( x ) := x | S | (1 + x ) F − P x i ∈ S f i and Ind( H ) denotes the set of all indep endent sets of H . Then deg C S = F − X x i ∈ S ( f i − 1) . Let R := { x i ∈ V ( H ) : f i = 1 } . Then deg C S = F if and only if S ⊆ R . Let H | R denote the induced subgraph of H on R . Hence the co efficient of x F in P G ( x ) (the n umber of indep enden t sets of size F ) is g F = | Ind( H | R ) | = P H | R (1) . 12 If P G ( x ) is symmetric, then g F = g 0 = 1. Since every graph on at least one v ertex has at least t wo indep enden t sets, namely ∅ and an y singleton, this forces R = ∅ . Th us f i ≥ 2 for all i . No w let T := { x i ∈ V ( H ) : f i = 2 } . Since f i ≥ 2 for all i , the only summand in ( 1 ) of degree F is the one corresp onding to S = ∅ , namely (1 + x ) F . Moreov er, deg C S = F − 1 if and only if S = { x i } for some x i ∈ T . Therefore the coefficient of x F − 1 in P G ( x ) receiv es con tributions only from the summand C ∅ ( x ) = (1 + x ) F and from the summands C { x i } ( x ) with x i ∈ T . The con tribution from C ∅ ( x ) is  F F − 1  = F . F or eac h x i ∈ T , we ha ve C { x i } ( x ) = x (1 + x ) F − 2 . Since there are | T | such v ertices, these terms con tribute a total of | T | . Th us g F − 1 = F + | T | . On the other hand, g 1 = | V ( G ) | = n + F . If P G ( x ) is symmetric, then g F − 1 = g 1 . So F + | T | = n + F . Hence | T | = n , whic h means f i = 2 for ev ery i . Con versely , if f i = 2 for ev ery i , then F = 2 n , and ( 1 ) b ecomes P G ( x ) = X S ∈ Ind( H ) x | S | (1 + x ) 2( n −| S | ) = (1 + x ) 2 n P H  x (1 + x ) 2  . Th us x 2 n P G (1 /x ) = x 2 n  1 + 1 x  2 n P H  1 /x (1 + 1 /x ) 2  = (1 + x ) 2 n P H  x (1 + x ) 2  = P G ( x ) . Therefore P G ( x ) is symmetric. □ 3.1. An application: caterpillar graphs. As an application of whisk ering, we no w fo cus on caterpillar graphs. Since a caterpillar is obtained from a path b y attaching lea v es along the central spine, the general results of this section translate in to explicit form ulas in terms of the p ositions of the legs. Encoding these p ositions by the gaps b etw een consecutiv e supp ort v ertices allows us to express P G ( − 1) as a pro duct of path contributions, from whic h the criteria for the v alues 0, 1, and − 1 follo w immediately . W e also provide a characterization of pseudo-Gorenstein ∗ caterpillars. Definition 3.3. A graph G is called a c aterpil lar if G is a tree and there exists a path x 1 , . . . , x n , called a c entr al p ath , such that every v ertex of G is either one of the x i ’s or is adjacen t to one of them. Ev ery v ertex of G that does not lie on the central path is called a le g . Notation 3.4. Let G b e a caterpillar with central path x 1 x 2 · · · x n . F or each i ∈ [ n ], let L i denote the set of all legs adjacen t to x i , i.e. L i := { v ∈ V ( G ) \ { x 1 , . . . , x n } : { v , x i } ∈ E ( G ) } . Set f i := | L i | . Th us G is obtained from P n b y attac hing f i lea ves to x i for each i ∈ [ n ]. Let B := { x i : f i > 0 } . Th us B records the p ositions on the cen tral path that carry at least one leg. 13 If B = ∅ , then G = P n . In this case, w e set r := 0 , m 0 := n and ℓ 0 := n . No w assume B  = ∅ and write B = { x b 1 < · · · < x b r : b 1 < · · · < b r } . The vertices of the central path carrying no legs form r +1 consecutive gaps. Let m 0 , m 1 , . . . , m r denote their lengths where: • m 0 is the n umber of vertices b efore x b 1 ; • m r is the n umber of vertices after x b r ; • for 1 ≤ j ≤ r − 1, m j is the n umber of vertices strictly betw een x b j and x b j +1 . Equiv alently , m 0 = b 1 − 1 , m j = b j +1 − b j − 1 (1 ≤ j ≤ r − 1) , m r = n − b r . Next, delete from the central path ev ery v ertex x b j together with its neigh b ors on the central path. The remaining vertices again form r + 1 path segments. Let ℓ 0 , ℓ 1 , . . . , ℓ r denote their lengths. Equiv alently , ℓ 0 = max { m 0 − 1 , 0 } , ℓ j = max { m j − 2 , 0 } (1 ≤ j ≤ r − 1) , ℓ r = max { m r − 1 , 0 } . Theorem 3.5. L et G b e a c aterpil lar as in Notation 3.4 . Then P G ( − 1) =        0 , if b j +1 = b j + 1 for some j, ( − 1) r r Y j =0 p ℓ j , otherwise , wher e p m := P P m ( − 1) is as in L emma 1.6 . Pr o of. Apply Prop osition 3.1 to the central path H = P n and recall that B = { x b 1 , . . . , x b r } . If t wo elements of B are consecutive, then B is not indep enden t in P n . Thus P G ( − 1) = 0 b y Prop osition 3.1 (a). Assume no w that no tw o elements of B are consecutiv e. Then B is an indep endent set of P n . By construction, deleting N P n [ B ] leav es exactly the r + 1 path segmen ts of lengths ℓ 0 , . . . , ℓ r . Hence P n − N P n [ B ] ∼ = r G j =0 P ℓ j . Therefore Prop osition 3.1 (b) and m ultiplicativity on disjoin t unions giv e P G ( − 1) = ( − 1) r r Y j =0 P P ℓ j ( − 1) = ( − 1) r r Y j =0 p ℓ j . □ Corollary 3.6. L et G b e a c aterpil lar as in Notation 3.4 . Then P G ( − 1) = 0 if and only if either b j +1 = b j + 1 for some j , or ℓ j ≡ 1 (mo d 3) for some j . Equivalently, a ( G ) = 0 if and only if no two vertic es of B ar e c onse cutive and ℓ j ≡ 1 (mo d 3) for al l j . 14 Pr o of. By Theorem 3.5 , the nontrivial case is when no tw o elements of B are consecutiv e, in which case P G ( − 1) = ( − 1) r r Y j =0 p ℓ j . No w Lemma 1.6 sa ys that p m = 0 if and only if m ≡ 1 (mo d 3). This pro v es the first assertion. The second statemen t follo ws from the fact a ( G ) = 0 whenev er P G ( − 1)  = 0. □ Corollary 3.7. L et G b e a c aterpil lar as in Notation 3.4 , and define λ :=    { j ∈ { 0 , . . . , r } : ℓ j ≡ 2 , 3 (mo d 6) }    . If P G ( − 1)  = 0 , then P G ( − 1) = ( − 1) r + λ . Pr o of. By Corollary 3.6 , the assumption P G ( − 1)  = 0 means that eac h p ℓ j ∈ {± 1 } . By Lemma 1.6 , one has p ℓ j = − 1 if and only if ℓ j ≡ 2 , 3 (mo d 6). Thus Q r j =0 p ℓ j = ( − 1) λ and Theorem 3.5 yields P G ( − 1) = ( − 1) r ( − 1) λ = ( − 1) r + λ . □ Corollary 3.8. L et G b e a c aterpil lar as in Notation 3.4 . Then α ( G ) = n X i =1 f i + r X j =0 l m j 2 m . Pr o of. By Prop osition 3.1 (c) with H = P n , we hav e α ( G ) = P n i =1 f i + α ( P n − B ). Since P n − B ∼ = F r j =0 P m j , we hav e α ( P n − B ) = r X j =0 α ( P m j ) = r X j =0 l m j 2 m . □ The following immediately follo ws from Corollaries 3.6 and 3.7 . Corollary 3.9. L et G b e a c aterpil lar as in Notation 3.4 . Then: (1) P G ( − 1) = 1 if and only if (a) no two c onse cutive vertic es of the c entr al p ath b oth c arry le gs; (b) ℓ j ≡ 1 (mo d 3) for every 0 ≤ j ≤ r ; and (c) r + λ is even. (2) P G ( − 1) = − 1 if and only if (a) no two c onse cutive vertic es of the c entr al p ath b oth c arry le gs; (b) ℓ j ≡ 1 (mo d 3) for every 0 ≤ j ≤ r ; and (c) r + λ is o dd. 15 Lastly , w e can conclude when a caterpillar is pseudo-Gorenstein ∗ b y applying Corollaries 3.7 and 3.8 . Corollary 3.10. L et G b e a c aterpil lar as in Notation 3.4 . Then G is pseudo-Gor enstein ∗ if and only if al l of the fol lowing hold: (1) no two vertic es of B ar e c onse cutive; (2) ℓ j ≡ 1 (mo d 3) for al l j = 0 , . . . , r ; (3) P x i ∈ B f i + P r j =0  m j 2  ≡ r + λ (mod 2) . W e conclude this section b y discussing when caterpillar graphs hav e symmetric indep endence p olynomials. The follo wing result follo ws immediately from Theorem 3.2 . Corollary 3.11. L et G b e a c aterpil lar as in Notation 3.4 such that B = { x 1 , . . . , x n } . Then P G ( x ) is symmetric if and only if f i = 2 for e ach i . 4. Independence pol ynomial of cochordal graphs In this section, we study indep endence p olynomials of co c hordal graphs. Using the quasi- forest structure of clique complexes of c hordal graphs and the asso ciated b -sequence from [ 10 ], w e classify the symmetric case, show that symmetry implies unimo dalit y , and record form ulas for P G ( − 1) and for the m ultiplicity of the ro ot − 1 in terms of graph in v ariants. W e first recall the definition of a quasi-forest. Definition 4.1. Let ∆ b e a simplicial complex with facets F 1 , . . . , F s . A facet F of ∆ is called a le af if either F is the unique facet of ∆, or there exists a facet B  = F of ∆ such that F ∩ H ⊆ F ∩ B for every facet H  = F of ∆. In this case, B is called a br anch of F . The simplicial complex ∆ is called a quasi-for est if its facets can b e ordered F 1 , . . . , F s in suc h a wa y that, for eac h i = 1 , . . . , s , the facet F i is a leaf of the subcomplex generated b y F 1 , . . . , F i . If, in addition, ∆ is connected, then ∆ is called a quasi-tr e e . An ordering F 1 , . . . , F s with the ab ov e property is called a le af or der . Next we recall the relationship betw een c hordal graphs and quasi-forests. Theorem 4.2. [ 9 , The or em 9.2.12] Given a finite simple gr aph G on n vertic es that is not c omplete, the fol lowing ar e e quivalent: (i) G is chor dal; (ii) the clique c omplex of G is a quasi-for est. W e in vestigate when the indep endence p olynomials of co c hordal graphs are symmetric. 16 Remark 4.3. Notice that when α ( G ) = 1, the indep endence p olynomial of G is P G ( x ) = 1 + | V ( G ) | x . So, for P G ( x ) to b e symmetric, w e need | V ( G ) | = 1. Th us symmetry forces G = K 1 . Theorem 4.4. L et G b e a c o chor dal gr aph and d = α ( G ) ≥ 2 . Then P G ( x ) is symmetric if and only if ther e exists an inte ger m ≥ 0 such that P G ( x ) = (1 + x ) d + mx (1 + x ) d − 2 . Mor e over, the indep endenc e p olynomial is unimo dal when it is symmetric. L astly, for every m ≥ 0 , the gr aph ( d − 2) K 1 ⊔ ( K m +2 − e ) is c o chor dal and has this indep endenc e p olynomial. Pr o of. Let ∆ b e the indep endence complex of G . Since G is co c hordal, G is chordal, and ∆ is the clique complex of G . Hence ∆ is a quasi-forest by [ 12 , Lemma 1.1]. In particular, dim ∆ = d − 1, and for 1 ≤ i ≤ d w e ha ve g i = f i − 1 (∆). By [ 10 , Theorem 1.1], there exist p ositiv e in tegers b 1 , . . . , b d suc h that d X i =1 g i ( x − 1) i − 1 = d X i =1 b i x i − 1 . Then (2) P G ( x ) = 1 + d X i =1 g i x i = 1 + x d X i =1 b i (1 + x ) i − 1 . F rom ( 2 ), the coefficients of x d , x d − 1 , and x in P G ( x ) are g d = b d , g d − 1 = b d − 1 + ( d − 1) b d , g 1 = d X i =1 b i . Assume that P G ( x ) is symmetric. Since g 0 = 1, symmetry giv es g d = g 0 = b d = 1. Symmetry also giv es g d − 1 = g 1 . So b d − 1 + ( d − 1) b d = d X i =1 b i . Since b d = 1, this b ecomes P d − 2 i =1 b i = d − 2. Since each b i is p ositive, it follo ws that b 1 = · · · = b d − 2 = 1. W rite b d − 1 = m + 1 with m ≥ 0. Substituting in to ( 2 ), we get P G ( x ) = 1 + x  d − 2 X i =1 (1 + x ) i − 1 + ( m + 1)(1 + x ) d − 2 + (1 + x ) d − 1  = 1 +  (1 + x ) d − 2 − 1  + ( m + 1) x (1 + x ) d − 2 + x (1 + x ) d − 1 = (1 + x ) d − 2  1 + ( m + 2) x + x 2  = (1 + x ) d + mx (1 + x ) d − 2 . 17 Notice that the p olynomial P G ( x ) = (1 + x ) d − 2  1 + ( m + 2) x + x 2  is unimo dal. T o see this, note that 1 + ( m + 2) x + x 2 has t wo real negativ e ro ots for m ≥ 0. Hence P G ( x ) is real-ro oted with nonnegative co efficients. Thus, it is log-concav e and therefore unimo dal. Con versely , supp ose that P G ( x ) = (1 + x ) d + mx (1 + x ) d − 2 for some m ≥ 0. Then P G ( x ) is symmetric as w e see b elow: x d P G (1 /x ) = x d  (1 + 1 /x ) d + m (1 /x )(1 + 1 /x ) d − 2  = (1 + x ) d + mx (1 + x ) d − 2 = P G ( x ) . Finally , consider the graph G obtained from deleting an edge, say e , from K m +2 . Then P G ( x ) = P K m +2 − e ( x ) = 1 + ( m + 2) x + x 2 . No w consider the graph H that is the disjoin t union of G and ( d − 2) isolated v ertices ( d − 2) K 1 . So H = ( d − 2) K 1 ⊔ ( K m +2 − e ). Hence P H ( x ) = (1 + x ) d − 2  1 + ( m + 2) x + x 2  = (1 + x ) d + mx (1 + x ) d − 2 . Moreo ver, G = K m +2 − e is an edge together with m isolated vertices. Hence chordal. Adding ( d − 2) isolated vertices on the G corresp onds to adding ( d − 2) vertices in the complemen t and they are connected to every v ertex in G , whic h preserves c hordalit y . There- fore H is co chordal and its indep endence p olynomial coincides with the desired symmetric form. □ Corollary 4.5. L et G b e a finite simple gr aph, and supp ose that G is chor dal with k c onne cte d c omp onents. Then P G ( − 1) = 1 − k . Pr o of. Set H := G . F or eac h i ≥ 1, let s ( K i , H ) denote the n umber of cliques of size i in H . Since indep endent sets of G are exactly cliques of H , we hav e P G ( x ) = 1 + X i ≥ 1 s ( K i , H ) x i . Th us, w e obtain P G ( − 1) = 1 + X i ≥ 1 ( − 1) i s ( K i , H ) . Since H is c hordal, it follows from [ 15 , Lemma 2] that cc( H ) = X i ≥ 1 ( − 1) i +1 s ( K i , H ) . Hence P G ( − 1) = 1 − cc( H ) = 1 − k . □ Remark 4.6. Corollary 4.5 is a direct reform ulation of [ 15 , Lemma 2]. As noted there, the clique count identit y for c hordal graphs already appears in [ 23 , Exercise 5.3.22]. In addition to the abov e nice corollary ab out the v alue of the indep endence p olynomial at x = − 1, there has b een in terest in finding the multiplicit y M ( G ) for co c hordal graphs. In particular, M ( G ) was expressed in terms of the graph structure in [ 5 ]. Belo w, we recall their result and add the relev an t implication ab out the degree of the h -p olynomial for co chordal graphs. Corollary 4.7. [ 5 , Cor ol lary 2.8] L et G b e a c o chor dal gr aph wher e G is r -c onne cte d. Then M ( G ) = r . As a r esult, the de gr e e of the h -p olynomial of the e dge ide al of G is deg h G ( t ) = α ( G ) − r . 18 5. V alues of the independence pol ynomial a t x = − 1 f or small α ( G ) In this section w e study the p ossible v alues of P G ( − 1) for connected graphs with α ( G ) ≤ 2. F or eac h n ≥ 3, w e consider R ( ≤ 2) n := { P G ( − 1) : G is a connected graph on n v ertices with α ( G ) ≤ 2 } . W e determine this set explicitly , show that ev ery v alue in it is realized b y a connected c hordal graph, and then observ e that | P G ( − 1) | can already grow exp onentially within the connected c hordal class. W e start with the following construction. Definition 5.1. Fix in teger n ≥ 3, 1 ≤ a ≤ ⌊ n/ 2 ⌋ , and 1 ≤ t ≤ n − a . Let A = { u 1 , . . . , u a } B = { v 1 , . . . , v n − a } b e disjoint sets. Let G n ( a, t ) denote the graph obtained from the disjoint union of the complete graphs on A and B b y adding the edges { u 1 , v j } for 1 ≤ j ≤ t . In other w ords, G n ( a, t ) is obtained from tw o complete graphs by joining one distinguished v ertex of the first clique to a t -clique of the second clique. Prop osition 5.2. The gr aph G n ( a, t ) is c onne cte d and chor dal. Mor e over, P G n ( a,t ) ( x ) = 1 + nx +  a ( n − a ) − t  x 2 . In p articular, P G n ( a,t ) ( − 1) = ( a − 1)( n − a − 1) − t . Pr o of. Set b := n − a . First, notice that G n ( a, t ) is connected since u 1 is adjacent to v 1 as t ≥ 1. Next, we prov e that G n ( a, t ) is chordal. F or this purp ose, w e sho w that u 2 , . . . , u a , v t +1 , . . . , v b , u 1 , v 1 , . . . , v t is a p erfect elimination ordering. Indeed, each v ertex u i with i ≥ 2 has all of its remaining neigh b ors inside the clique A , and each vertex v j with j > t has all of its remaining neigh b ors inside the clique B . After deleting these v ertices, the remaining graph on { u 1 , v 1 , . . . , v t } is the clique K t +1 . Hence G n ( a, t ) is chordal. Since b oth A and B are cliques, every indep enden t set of G n ( a, t ) has size at most 2. Th us P G n ( a,t ) ( x ) = 1 + nx + g 2 x 2 , where g 2 is the n um b er of independent sets of size 2. Any such pair must consist of one v ertex of A and one vertex of B . There are ab = a ( n − a ) suc h pairs in total, and exactly t of them, namely { u 1 , v j } for 1 ≤ j ≤ t , are edges. Therefore g 2 = a ( n − a ) − t . Hence P G n ( a,t ) ( x ) = 1 + nx +  a ( n − a ) − t  x 2 . Therefore, P G n ( a,t ) ( − 1) = 1 − n + a ( n − a ) − t = ( a − 1)( n − a − 1) − t. □ 19 Remark 5.3. Notice that if α ( G ) ≤ 2, then the complement G is triangle-free. There is a w ell-known b ound on the n umber of edges of such a graph due to Mantel. It states that the maxim um n umber of edges in a triangle-free graph on n vertices is j n 2 4 k . Equalit y holds if and only if the graph is K ⌊ n 2 ⌋ , ⌈ n 2 ⌉ . W e obtain the follo wing description utilizing the mentioned result of Man tel from ab ov e. Theorem 5.4. L et n ≥ 3 . Then R ( ≤ 2) n =  − ( n − 1) ,  ( n − 2) 2 4  − 1  ∩ Z wher e R ( ≤ 2) n = { P G ( − 1) : G is a c onne cte d gr aph on n vertic es with α ( G ) ≤ 2 } . Pr o of. Let G b e a connected graph on n v ertices with α ( G ) ≤ 2. Then ev ery indep endent set has size at most 2. So P G ( x ) = 1 + nx + g 2 x 2 , where g 2 is the num b er of indep enden t sets of size 2. These indep enden t pairs are exactly the edges of G . Hence g 2 = e ( G ). Therefore, P G ( − 1) = 1 − n + e ( G ) . Since α ( G ) ≤ 2, the complemen t G is triangle-free. Hence, by Mantel’s theorem, e ( G ) ≤  n 2 4  . Moreo ver, equalit y cannot occur b ecause then G w ould b e complete bipartite, and so G w ould b e the disjoin t union of t wo cliques, con tradicting the connectedness of G . Th us e ( G ) ≤  n 2 4  − 1 . It follows that P G ( − 1) ≤ 1 − n +  n 2 4  − 1  =  ( n − 2) 2 4  − 1 . On the other hand, e ( G ) ≥ 0, so P G ( − 1) ≥ 1 − n = − ( n − 1) . Hence R ( ≤ 2) n ⊆  − ( n − 1) ,  ( n − 2) 2 4  − 1  ∩ Z . F or the reverse inclusion, fix a with 1 ≤ a ≤ ⌊ n/ 2 ⌋ , and let I a := [( a − 1)( n − a − 1) − ( n − a ) , ( a − 1)( n − a − 1) − 1] ∩ Z . By Prop osition 5.2 , as t runs through 1 , . . . , n − a , the graphs G n ( a, t ) realize exactly the in tegers in I a . Notice that I 1 = [ − ( n − 1) , − 1] ∩ Z . If U a denotes the righ t endp oint of I a and L a +1 the left endp oin t of I a +1 , then U a − L a +1 =  ( a − 1)( n − a − 1) − 1  −  a ( n − a − 2) − ( n − a − 1)  = a − 1 ≥ 0 . 20 Notice that I 1 ∪ I 2 ∪ · · · ∪ I ⌊ n/ 2 ⌋ is a single in terv al of integers since I a ∩ I a +1  = ∅ for ev ery 1 ≤ a < ⌊ n/ 2 ⌋ . Then its largest endp oin t is attained at a = ⌊ n/ 2 ⌋ as ( a − 1)( n − a − 1) is increasing for 1 ≤ a ≤ ⌊ n/ 2 ⌋ . Therefore max( I 1 ∪ I 2 ∪ · · · ∪ I ⌊ n/ 2 ⌋ ) = j n 2 k − 1  l n 2 m − 1  − 1 =  ( n − 2) 2 4  − 1 . Th us I 1 ∪ I 2 ∪ · · · ∪ I ⌊ n/ 2 ⌋ =  − ( n − 1) ,  ( n − 2) 2 4  − 1  ∩ Z . Since eac h G n ( a, t ) is connected, c hordal, and satisfies α ( G n ( a, t )) ≤ 2, the rev erse inclusion follo ws. □ Corollary 5.5. Every inte ger o c curs as P G ( − 1) for some c onne cte d chor dal gr aph G . In fact, [ n ≥ 3 { P G ( − 1) : G c onne cte d chor dal on n vertic es with α ( G ) ≤ 2 } = Z . Pr o of. By Theorem 5.4 , for eac h n ≥ 3 w e obtain the full interv al  − ( n − 1) ,  ( n − 2) 2 4  − 1  ∩ Z . As n v aries, the left endp oints tend to −∞ and the righ t endp oints tend to + ∞ . Hence ev ery in teger is realized. □ Next we show that | P G ( − 1) | can b e muc h larger in the general connected chordal case. Definition 5.6. Let q ≥ 1 and let r 1 , . . . , r q b e positive in tegers. The clique b ouquet H ( r 1 , . . . , r q ) is the graph obtained from the cliques K r 1 +1 , . . . , K r q +1 b y identifying one chosen vertex of eac h clique to a single common vertex x . Prop osition 5.7. The gr aph H ( r 1 , . . . , r q ) is c onne cte d and chor dal. Mor e over, P H ( r 1 ,...,r q ) ( x ) = x + q Y i =1 (1 + r i x ) and P H ( r 1 ,...,r q ) ( − 1) = Q q i =1 (1 − r i ) − 1 . Pr o of. The graph is connected since ev ery vertex is adjacent to the common v ertex x . It is c hordal b ecause every blo ck is a clique. An indep endent set of H ( r 1 , . . . , r q ) either con tains x , in which case it con tributes the term x , or it a voids x . In the latter case, from eac h clique K r i +1 \ { x } one ma y choose either no v ertex or exactly one vertex. Hence the con tribution from the i th clique is 1 + r i x . These c hoices are indep endent across i . Therefore P H ( r 1 ,...,r q ) ( x ) = x + q Y i =1 (1 + r i x ) , and ev aluating at x = − 1 yields the form ula. □ 21 The next theorem shows that the extremal b ehavior in the full connected chordal class is exp onen tial rather than quadratic. Theorem 5.8. L et M n := max {| P G ( − 1) | : G is a c onne cte d chor dal gr aph on n vertic es } . Then for every n ≥ 3 , M n ≥ 4 ⌊ n − 1 5 ⌋ − 1 . In p articular, M n gr ows exp onential ly with n . Pr o of. F or n ≤ 5, the stated bound is trivial. Assume n ≥ 6, and write n − 1 = 5 q + r with 0 ≤ r ≤ 4. W e construct a clique b ouquet on n v ertices as follows:                                    H (5 , . . . , 5 | {z } q ) if r = 0 , H (5 , . . . , 5 | {z } q − 1 , 3 , 3) if r = 1 , H (5 , . . . , 5 | {z } q , 2) if r = 2 , H (5 , . . . , 5 | {z } q , 3) if r = 3 , H (5 , . . . , 5 | {z } q , 4) if r = 4 . In eac h case the sum of the displa yed parameters is n − 1, so the graph has n v ertices. It follo ws from Prop osition 5.7 that P G ( − 1) = Y i (1 − r i ) − 1 . Moreo ver, in the fiv e cases r = 0 , 1 , 2 , 3 , 4, w e hav e      Y i (1 − r i )      = 4 q , 4 q , 4 q , 2 · 4 q , 3 · 4 q . Therefore | P G ( − 1) | ≥ 4 q − 1 = 4 ⌊ n − 1 5 ⌋ − 1 . □ Remark 5.9. It w as pro ved by Engstr¨ om in [ 4 , Corollary 3.2] that | P G ( − 1) | ≤ 2 φ ( G ) where φ ( G ) denotes the decycling n umber of G . The decycling n umber is the minim um num b er of v ertices whose deletion from G turns it in to a forest. Note that 4 ⌊ n − 1 5 ⌋ − 1 = 2 2 ⌊ n − 1 5 ⌋ − 1 . Since, for n ≥ 3, 2  n − 1 5  ≤ 2 n − 2 5 ≤ n − 2 , it follows that 4 ⌊ n − 1 5 ⌋ − 1 ≤ 2 n − 2 − 1 < 2 n − 2 . 22 On the other hand, if G is any connected graph on n ≥ 2 ve rtices, then φ ( G ) ≤ n − 2. Indeed, since G is connected, it con tains an edge uv , and deleting all v ertices other than u and v lea ves the graph K 2 , which is acyclic. Hence the general b ound | P G ( − 1) | ≤ 2 φ ( G ) yields | P G ( − 1) | ≤ 2 n − 2 . Therefore, with M n := max {| P G ( − 1) | : G is a connected c hordal graph on n vertices } , w e obtain 4 ⌊ n − 1 5 ⌋ − 1 ≤ M n ≤ 2 n − 2 . 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Kara) Dep ar tment of Ma thema tics, Br yn Ma wr College, Br yn Ma wr, P A 19010 Email addr ess : skara@brynmawr.edu (D. Vien) Dep ar tment of Ma thema tics, Br yn Ma wr College, Br yn Ma wr, P A 19010 Email addr ess : dvien@brynmawr.edu