Which Digraphs with Ring Structure are Essentially Cyclic?

Whic h Digraphs with Ring Structure are Essen ti ally Cyclic? Rafig Agaev 1 and P a v el Chebo tarev 2 Institute of Control Sciences of the Russian Academ y of Sciences 65 Profsoyuznay a Street, Moscow 11 7997 , Russia Abstract W e sa y that a digraph is essen tially cyclic if its Laplacian sp ectrum is not completely real. The essen tial cy clicit y implies the presence of directed cycles, b ut not vice ve rsa. The problem of characteri zing ess ential cyclicit y in terms of graph top ology is difficult and yet unsolved. It s solution is important for some applications of graph t heory , including that in decentralized contro l. In the present pap er, this problem is solved with resp ect to th e class of digraphs with ring structure, which mod els some typical comm unication net wor k s. It is shown that th e digraphs in this class are essen tially cy clic, except for certain sp ecified digraphs. The main technical to ol we employ is th e Cheb yshev p olynomials of the second k ind. A by-pro duct of this study is a theorem on the zeros of p olynomials that differ by one from the p rodu cts of Chebyshev p olynomials of th e second kind. W e also consider the p roblem of essen tial cy clicit y for weigh ted d igraphs and enumerate the spannin g trees in some digraphs with ring structure. Keywor ds : Laplacian matrix; Laplacia n sp ectrum; Essential cyclicity; Chebyshev polynomial; Spanning tree; Directed graph; W eigh ted digraph 1. In tro duc tion As distinct from the Laplacian eigenv alues of ordina ry gra phs, those of digr aphs need not b e re a l. The problem of characterizing all digraphs that have completely re al La placian sp ectra is difficult a nd yet unsolved. Obviously , tw o class es of s uch digraphs are: the acyclic digraphs, whos e Laplacian matrices hav e a tria ng ular form, and symmetr ic digraphs, whose Laplac ian matrices a re symmetric and p ositive semidefinite. It can also b e obser ved that the sp ectrum of a Laplacian matrix L is completely r eal whenever for so me ε > 0, I − εL is the trans ition matrix of a reversible Mar ko v chain. The prev io us results [3] suggest that non-real eig env alues with noticeable imaginary pa rts characterize digraphs that hav e directed c y cles not “extinguished” by counter-directional cycles. On the other hand, there are digraphs that hav e “undamped” cycles and completely r e al sp ectra . In general, it is not an easy problem to distinguish, in terms of graph top olog y , dig raphs with rea l Laplacian sp e ctra fro m those having some non-rea l Laplacia n eigenv alues. The digraphs of the second type are guar anteed to hav e cycles, and we call them essential ly cyclic. Some pr e liminary results on essentially cyclic w eig hted digra phs were presented in [7, 9]. The afor ementioned pr oblem of distinguishing, in terms of graph top olog y , digr aphs w ith rea l and partially no n-real sp ectra is impo rtant for applications. In par ticular, in the decen tralized c o ntrol of m ulti-a g ent systems [10, 19, 21, 23], the abse nce of non-rea l Lapla cian e igenv alues o f the communication digraph implies that the simplest cons e ns us algorithms are dev oid of os cillations. 1 E-mail: ar p o@ipu.r u 2 Corresp onding auth or. E-m ail: ch v@member.ams.org, upi@ipu.ru; phone : +7-495-334-8869; fax: +7-495-420-2016 . 2 A ratio nal appro a ch to attacking the difficult problem of characterizing esse ntially cyclic directed graphs is studying v ario us classes of cyclic digraphs. In this pa per , we inv estigate the digr aphs with ring structure . By such a digraph we mean a digraph whose ar c set only co nt ains a collection of a rcs forming a Ha miltonian cyc le and an arbitra ry num b er o f arcs tha t b elong to the inverse Hamiltonian cycle. Digraphs of this type mo del a class of typical asymmetr ic co mmunication netw orks. W e obtain a necessary and s ufficien t condition of es sential c y clicity for the digraphs with ring structure. According to this condition, such digraphs are essentially cyclic, except for the digraphs whose inverse Hamiltonian cycle lacks tw o mos t dis tant ar cs o r one arc or no arc. This study in volves the Chebyshev p olyno mia ls of the second kind. As a by-pro duct w e o bta in a theorem on the zero s of p o lynomials that differ by one from the pro ducts of Chebyshev p oly nomials of the second kind. W e also consider weighted digra phs and find that in this case, some conditions o f essential c yclicity inv o lve the tria ngle inequality for the square ro o ts of weigh t differences. Finally , we e n umera te the conv erging trees in some digraphs with ring structure. The pap er is o rganized as follows. After the necessar y nota tion and pr eliminary results (Section 2), in Section 3 we present three auxiliary lemmas needed to pr ov e the main results. In Sectio n 4, we obtain a necessa ry and sufficien t conditio n of essential cyclicit y for the dig raphs with r ing structure, i. e., for the digraphs consisting of t wo oppo site Hamiltonian cycles from one of which some arcs ca n b e removed. In Section 5, w e in vestigate the essent ia l cyclicit y of the simplest weigh ted digr aphs. Finally , in Section 6 , we present explicit formulas for the num b er o f conv erging trees (in-ar bo rescences) in certain digraphs with ring structure and a dir ect computation, by means of Chebyshev polynomials , of the Laplacia n sp ectrum of the undirected cycle, which is usually prov ed rather than derived. 2. Notation and basic results The Laplacian matrix o f a digra ph Γ with v ertex set V (Γ) = { 1 , . . . , n } and a rc s et E (Γ) is the ma trix L = ( ℓ ij ) ∈ I R n × n in which, for j 6 = i , ℓ ij = − 1 whenev er ( i, j ) ∈ E (Γ), otherwise ℓ ij = 0; ℓ ii = − P j 6 = i ℓ ij , i, j ∈ V (Γ). If a digraph Γ is w eighted, i. e., each ar c ( i, j ) ∈ E (Γ) has a strictly positive weigh t w ij , then in the Lapla cian ma trix L = L (Γ) = ( ℓ ij ), for j 6 = i , ℓ ij = − w ij whenever ( i, j ) ∈ E (Γ), other wise ℓ ij = 0 ; ℓ ii = − P j 6 = i ℓ ij , i , j ∈ V (Γ). Among the pa p er s concerned with the Laplacian ma trices of digraphs, we men tion [1–3, 6, 8, 1 1 , 12 , 16, 24] and [1 3], whe r e the Laplacia n matr ix is defined differen tly . W e say that a (weigh ted) digra ph Γ is essential ly cyclic if its Laplacian sp ectrum contains non-r eal eigenv alues. Evidently , ev er y essentially cyclic digraph has at least o ne dir ected cycle. Indeed, otherwise there exists a n umbering of vertices such that the L a placian matrix has a tr iangular form, so the Laplacia n sp ectrum consists of the real diag onal en tries of this form. The Chebyshev p olynomial of t he se c ond kind , P n ( x ) , sc ale d on ] − 2 , 2 [ is the po lynomial of degree n defined by P n ( x ) = sin(( n + 1) arccos x 2 ) q 1 − x 2 4 , where x ∈ ] − 2 , 2[ . (1) Using the auxiliary v aria ble ϕ ∈ ]0 , π [ s uch that x = 2 cos ϕ (i. e., ϕ = arccos x 2 ) one can rewrite (1) in the form P n ( x ) = sin(( n + 1) ϕ ) sin ϕ . (2) The explicit form of the p olynomials P n ( x ) is (see, e.g., Theorem 1.1 2 in [20]) 3 P n ( x ) = [ n/ 2] X i =0 ( − 1) i  n − i i  x n − 2 i (3) and they satisfy the recurr ence P n ( x ) = xP n − 1 ( x ) − P n − 2 ( x ) (4) with the initial conditions P 0 ( x ) ≡ 1 and P 1 ( x ) ≡ x . By (2), the ro o ts o f P n ( x ) are: x k = 2 cos π k n + 1 , k = 1 , . . . , n . (5) In particular, if n = 2 m − 1 a nd m > 0 is integer, then P n ( x ) = 2 m − 1 Y k =1  x − 2 cos π k 2 m  = x m − 1 Y k =1  x − 2 cos π k 2 m  x + 2 cos π k 2 m  = x m − 1 Y k =1  x 2 − 4 c o s 2 π k 2 m  . (6) Consider the tridiagonal matrix M n ∈ I R n × n : M n =              2 − 1 0 − 1 2 − 1 . . . . . . . . . . . . . . . . . . − 1 2 − 1 0 − 1 1              . (7) In par ticular, M 1 =  1  . Let Z n ( x ) b e the characteristic po lynomial of M n : Z n ( x ) = det( xI − M n ). The expansion along the first row of xI − M n for every n ≥ 2 pr ovides Z n ( x ) = ( x − 2) Z n − 1 ( x ) − Z n − 2 ( x ) , (8) with the initia l conditions Z 0 ( x ) ≡ 1 a nd Z 1 ( x ) ≡ x − 1. The p olynomials Z n ( x ) play a ce n tra l ro le in the subsequent cons ider ations. 3. Auxiliary lemmas It will b e shown in Sectio n 4 that the Laplacian characteristic p oly nomials of the digra phs with ring structure differ by o ne from the pro ducts of the p olynomials Z n ( x ). In this section, we prove three lemmas. Le mma 1 connec ts Z n ( x ) with the Chebyshev polynomia ls P n ( x ) of the second kind, Lemma 2 provides the explicit for m of Z n ( x ), and Lemma 3 spe c ifie s the ro ots of the p olynomials Z n ( x ) ± 1. Lemma 1. F or n = 0 , 1 , 2 , . . . , Z n ( x 2 ) ≡ P 2 n ( x ) . Pro of. The pro of pro ceeds by inductio n. F or n = 0, b y definition, Z 0 ( x 2 ) ≡ P 0 ( x ) ≡ 1 holds. F or n = 1 we hav e Z 1 ( x 2 ) = x 2 − 1 = P 2 ( x ). Assume that the requir ed statement holds for n = m − 1 a nd n = m and show that it is true for n = m + 1. Indeed, b y (4) and (8), Z m +1 ( x 2 ) = ( x 2 − 2) Z m ( x 2 ) − Z m − 1 ( x 2 ) = ( x 2 − 2) P 2 m ( x ) − P 2 m − 2 ( x ) = x ( xP 2 m ( x ) − P 2 m − 1 ( x )) + ( xP 2 m − 1 ( x ) − P 2 m − 2 ( x )) − 2 P 2 m ( x ) = xP 2 m +1 ( x ) + P 2 m ( x ) − 2 P 2 m ( x ) = P 2 m +2 ( x ). ⊓ ⊔ Lemma 2. 1 . The explicit form of the p olynomial Z n ( x ) is 4 Z n ( x ) = n X i =0 ( − 1) i  2 n − i i  x n − i . (9) 2 . The s et of r o ots of Z n ( x ) is n 4 co s 2 π k 2 n +1    k = 1 , . . . , n o and t hey al l b elong to [0 , 4[ . Pro of. Due to Lemma 1, to verify item 1, it suffices to compa r e (9) with (3); item 2 follo ws from (5). ⊓ ⊔ Since for x ≥ 0 P 2 n ( √ x ) = Z n ( x ), (1) provides the following trigono metric r epresentation of Z n ( x ): Z n ( x ) = sin((2 n + 1) ϕ ) sin ϕ , where x = 4 cos 2 ϕ, ϕ ∈  0 , π 2  , x ∈ [0 , 4[ . (10) Lemma 3. The set of the r o ots of the e quation Z n ( x ) + ( − 1) p = 0 , p ∈ { 0 , 1 } (11) is n 4 co s 2 π k 2 n +1+( − 1) k + p    k = 1 , . . . , n o and t hey al l b elong to [0 , 4 [ . Pro of. By (10) the ro o ts x i of Eq. (11) are connected with the ro o ts ϕ i of the equation sin((2 n + 1) ϕ ) sin ϕ + ( − 1) p = 0 (12) by x i = 4 cos 2 ϕ i . W e first solve Eq. (1 2) and then return to (11). O bserve that if (2 n + 1) ϕ = πk − ( − 1) k + p ϕ, (13) where k is integer, then the equality sin((2 n + 1) ϕ ) = ( − 1) p +1 sin ϕ to which Eq. (12) reduces w hen ϕ ∈  0 , π 2  is satisfied. By (13), ϕ = π k 2 n + 1 + ( − 1) k + p . (14) T aking (14) with k = 1 , . . . , n provides n distinct roo ts of Eq. (12); all o f them belo ng to ]0 , π 2 ]. Due to (10), the set of the corresp onding distinct roots of Eq. (1 1 ) in [0 , 4[ is { 4 cos 2 π k 2 n +1+( − 1) k + p   k = 1 , . . . , n } . Since the degree of (11 ) is n , this equation has no other ro o ts. ⊓ ⊔ 4. Essen tially cyclic digraphs with ring structure By a digraph with r ing structure we mean a dig raph tha t contains a Hamiltonia n cycle and whose remaining arcs b elong to the inv ers e Hamiltonian cycle. More s pec ifically , let Γ 1 n = ( V n , E 1 n ) a nd Γ 2 n = ( V n , E 2 n ) b e the digraphs with V n = { 1 , . . . , n } , E 1 n = { (1 , n ) , ( n, n − 1 ) , . . . , (2 , 1) } , and E 2 n = E 1 n ∪ { (1 , 2) , (2 , 3) , . . . , ( n − 1 , n ) , ( n, 1) } . W e say that Γ n = ( V n , E ) is a di gr aph with ring stru ctur e if it is isomorphic to some e Γ n = ( V n , e E ) with E 1 n ⊆ e E ⊆ E 2 n . Digraphs Γ 1 n and Γ 2 n are shown in Fig. 1 (a ) and 1(b), resp ectively . In this sec tion, w e answ er the question in the title of the pap er and find the Laplacia n sp ectra of some digraphs with ring structure. Certain weighte d digr aphs with ring structure are considered in Section 5 . 4.1. The digraphs Γ 1 n with n arcs and Γ 2 n with 2 n arcs Theorem 1. 1. Γ 1 n is ess en tial ly cyclic ; its La placian sp e ct r u m is n 2 sin 2 π k n + i sin 2 π k n    k = 1 , . . . , n o . 2. Γ 2 n is n ot essential ly cyclic ; its L aplacian sp e ctru m is n 4 sin 2 π k 2 n    k = 1 , . . . , n o . 5 ❄ (a) q ❥ ❥ ❨ ❨ ❨ ☛ ✕ 1 2 3 4 5 n n − 1 ✛ ✛ ✻ ✛ (b) ✲ ✲ ✲ q ❥ ❥ ❨ ❨ ❨ ☛ ✕ 1 2 3 4 5 n n − 1 ✛ ✛ ✻ ✛ (c) ✲ ✲ ✲ q ❥ ❥ ❨ ❨ ❨ ☛ ✕ 1 2 3 4 5 n n − 1 Figure 1. The digraphs: ( a) Γ 1 n ; (b) Γ 2 n ; (c) Γ ′ n . Pro of. 1. The La placian matrix of Γ 1 n is L 1 n = I − Q , where Q = ( q ij ) ∈ I R n × n is the circulant per mut atio n matrix with the entries q ij = 1 whenever i − j ∈ { 1 , 1 − n } and q ij = 0 other wise. The characteristic poly nomial o f Q is ∆ Q ( λ ) = λ n − 1 and its sp ectrum is  e − 2 π k i /n   k = 1 , . . . , n  . Therefore the eigenv alues of L 1 n = I − Q are λ k = 1 − e − 2 π k i /n = 1 − cos 2 π k n + i sin 2 π k n = 2 sin 2 π k n + i sin 2 π k n , k = 1 , . . . , n (cf. § 2.1 in [1 4 ], Section 4.8.3 in [18], and Section 4 in [3]). 2. The Laplacian matrix of Γ 2 n is sy mmetr ic and co incides with that o f the undirected n - cycle. The sp ectrum of this matrix w as found in [15] and [4] and coincides with the expression given in Theorem 1. On the deriv a tion of this expressio n, see Section 6 . ⊓ ⊔ The following repr esentation of the sp ectrum of Γ 1 n is a consequence of Theorem 1. Corollary of Theorem 1. The La placian sp e ctrum of Γ 1 n is n 2 sin π k n e π i ( 1 2 − k n )    k = 1 , . . . , n o . 4.2. The digraphs Γ ′ n with 2 n − 1 arcs Consider the digraph that differ s from Γ 2 n by one arc. L e t Γ ′ n = ( V n , E ′ n ), where E ′ n = E 2 n r { ( n, 1) } , see Fig. 1(c). The La placian matrix of Γ ′ n , L ′ n =              2 − 1 0 · · · 0 − 1 − 1 2 − 1 0 . . . . . . . . . . . . . . . . . . . . . 0 − 1 2 − 1 0 − 1 1              , differs from the tridia gonal matrix M n (see (7)) by the non-zero (1 , n ) entry . Theo rem 2 b elow states that the Lapla cian characteristic p olynomia l of Γ ′ n can b e express ed via the polyno mial Z n int ro duced in Section 1 and that Γ ′ n , as well as Γ 2 n , is not essentially cyclic. Theorem 2 . L et L ′ n b e t he L aplacian matrix of digr aph Γ ′ n whose ar cs c onst itu te the Hamiltonian cycle (1 , n ) , ( n, n − 1) , . . . , (2 , 1) and the p ath (1 , 2) , (2 , 3 ) , . . . , ( n − 1 , n ) , and ther e ar e no other ar cs. Then : 1. The char acteristic p olynomial of L ′ n is ∆ L ′ n ( λ ) = Z n ( λ ) − ( − 1) n . 2. Γ ′ n is n ot essential ly cyclic and its L aplacian sp e ctrum is { 4 co s 2 π k 2 n +1 − ( − 1) k + n , k = 1 , . . . , n } . Pro of. 1. Expanding det( λI − L ′ n ) along the first row and making use of (8) provide ∆ L ′ n ( λ ) = det( λI − L ′ n ) = ( λ − 2 ) Z n − 1 ( λ ) − Z n − 2 ( λ ) − ( − 1 ) n = Z n ( λ ) − ( − 1 ) n . 2. By Lemma 3, the ro ots of ∆ L ′ n ( λ ) a r e 4 cos 2 π k 2 n +1 − ( − 1) k + n , k = 1 , . . . , n , s o Γ ′ n is no t essentially cyclic. ⊓ ⊔ 6 4.3. The digraphs Γ ′′ n with 2 n − 2 arcs Now cons ider the dig raphs with ring structure c onsisting of a Hamiltonian cycle supplemented b y the inv e rse Hamiltonian cycle in w hich two ar bitrary arcs are lacking. A digraph Γ ′′ n of this kind r esults from Γ ′ n by removing some ( i, i + 1) arc, 1 ≤ i < n . Therefore the Laplacia n matrix L ′′ n of Γ ′′ n is o btained from L ′ n by changing t wo ele men ts in the i th r ow: L ′′ n =                        2 − 1 0 · · · · · · · · · · · · 0 − 1 − 1 2 − 1 0 . . . . . . . . . . . . . . . . . . . . . . . . − 1 1 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 − 1 2 − 1 0 − 1 1                        1 2 . . . . . . i . . . . . . n − 1 n (15) It turns out tha t a nswering the question of whether Γ ′′ n is essentially cyclic reduces to studying the pro ducts of Chebyshev p oly nomials of the second kind. T o do this, we need the following notation. Let x ( m ) 1 = 4 cos 2 π m 2 m + 1 and x ( m ) 2 = 4 cos 2 π ( m − 1 ) 2 m + 1 (provided that m > 1) (16) be the smallest and the seco nd smallest ro ots of Z m ( x ), resp ectively (see Le mma 2); u ( m ) 1 = 4 cos 2 π m 2( m + 1) and u ( m ) 2 = 4 cos 2 π ( m − 1 ) 2 m (provided that m > 1) (17) are the smallest and the second smallest ro o ts of Z m ( x ) + ( − 1 ) m , resp ectively (see Lemma 3). ✲ ✻ 1 − 1 x ( j ) 1 x ( j ) 2 x ( i ) 1 x ( i ) 2 u ( j ) 1 u ( j ) 2 u ( i ) 1 u ( i ) 2 Z i Z j Z i Z j Figure 2. The shapes of th e p olynomials Z i ( x ), Z j ( x ), and Z i ( x ) Z j ( x ) when i and j are o dd and 0 < i < j − 1. The following lemma es ta blishes s everal inequa lities in volving the roo ts (16)–(17) o f Z m ( x ) and Z m ( x ) + ( − 1 ) m as well as the p olynomials that have the pro duct fo rm Z i ( x ) Z j ( x ). The shap es o f Z i ( x ), Z j ( x ), and Z i ( x ) Z j ( x ) when i and j are o dd are exemplified in Fig. 2. Lemma 4. L et 0 < i < j − 1 . 1. If u ( i ) 1 < x ( j ) 2 , then i j − 1 > 2 3 . 7 2. u ( i ) 1 > u ( j ) 2 . 3. The ine quality | Z i ( x ) Z j ( x ) | < 1 (18) holds for al l x ∈ ]0 , ma x( x ( i ) 1 , u ( j ) 2 )] . 4. If u ( i ) 1 < x ( j ) 2 , then (1 8 ) is satisfie d for al l x ∈ [ u ( i ) 1 , x ( j ) 2 ] . Lemma 4 is the main technical tool employ ed in the pro o fs of the subsequent theor ems of this section. Pro of. 1 . Since c os 2 t strictly decreases on [0 , π 2 ], u ( i ) 1 < x ( j ) 2 implies that i 2( i +1) > j − 1 2 j +1 . As a consequence, one has item 1 of Lemma 4. Item 2 follows from the definitions of u ( i ) 1 and u ( j ) 2 and the inequality 0 < i < j − 1. 3. Let x ( i ) 1 ≥ u ( j ) 2 . Let us show that (18) holds for all x ∈ ]0 , x ( i ) 1 ]. If x ∈ ]0 , x ( i ) 1 [, then ϕ ∈ ] π i 2 i +1 , π 2 [, where ϕ = arc c os x 2 . Let ϕ = π ( i + δ ) 2 i +1 for 0 < δ < 1 / 2. W e get | Z i ( x ) Z j ( x ) | ≤    sin(2 i + 1) ϕ sin 2 ϕ    =      sin π δ sin 2 π ( i + δ ) 2 i +1      . (19) Consider the deriv ative of sin δ π  sin 2 π ( i + δ ) 2 i +1 : sin π δ sin 2 π ( i + δ ) 2 i +1 ! ′ δ = π cos( π δ ) sin π ( i + δ ) 2 i +1 − 2 π 2 i +1 sin( π δ ) c o s π ( i + δ ) 2 i +1 sin 3 π ( i + δ ) 2 i +1 =  π (2 i − 1) 2 i +1 + 2 π 2 i +1  cos( δ π ) sin π ( i + δ ) 2 i +1 − 2 π 2 i +1 sin( δ π ) c o s π ( i + δ ) 2 i +1 sin 3 π ( i + δ ) 2 i +1 = π (2 i − 1) 2 i +1 cos( π δ ) sin π ( i + δ ) 2 i +1 + 2 π 2 i +1 sin π i (1 − 2 δ ) 2 i +1 sin 3 π ( i + δ ) 2 i +1 . (20) Since for 0 ≤ δ < 1 / 2 the trig o nometric functions in (20) ar e pos itive, so is the le ft-ha nd side, therefore sin π δ  sin 2 π ( i + δ ) 2 i +1 increases in δ . F or δ = 0 and δ = 1 / 2, sin π δ  sin 2 π ( i + δ ) 2 i +1 equals 0 and 1 , res pec tively . Hence sin π δ  sin 2 π ( i + δ ) 2 i +1 < 1 holds for all δ ∈ [0 , 1 2 [ and, b y (19), | Z i ( x ) Z j ( x ) | < 1 is sa tisfied for all x ∈ ]0 , x ( i ) 1 ]. Let us s how that u ( j ) 2 > x ( i ) 1 implies | Z i ( x ) Z j ( x ) | < 1 as w ell. Cons ider the para metrization ϕ = ϕ ( δ ) = π 2 j − δ j , from which x = x ( δ ) = 4 co s 2 π 2 j − δ j . Then w e obtain x (0) = 0 , x  j j +1  = u ( j ) 1 , and x (1) = u ( j ) 2 . F or x ∈ ]0 , u ( j ) 2 ] (i. e., for δ ∈ ]0 , 1]) w e hav e Z i ( x ) Z j ( x ) = sin  π 2 (2 i + 1) j − δ j  sin  π 2 (2 j + 1) j − δ j  sin 2  π 2 j − δ j  , consequently , | Z i ( x ) Z j ( x ) | ≤   sin  π 2 (2 i + 1)  1 − δ j     sin 2  π 2  1 − δ j   =   cos  δπ 2 2 i +1 j    cos 2  δπ 2 1 j  . If cos  δπ 2 2 i +1 j  > 0 , then δπ 2 2 i +1 j < π 2 , so | Z i ( x ) Z j ( x ) | ≤ cos  δπ 2 2 i +1 j  cos 2  δπ 2 1 j  < cos  δπ 2 2 i j  cos 2  δπ 2 1 j  = cos 2  δπ 2 i j  − sin 2  δπ 2 i j  cos 2  δπ 2 1 j  < 1 . If cos  δπ 2 2 i +1 j  ≤ 0 , then π 2 ≤ δπ 2 2 i +1 j < π , a nd since i + 1 < j , we hav e π 2 ≤ π 2 2 i +1 j < π . Therefore 8   cos  δπ 2 2 i +1 j    cos 2  δπ 2 1 j  ≤   cos  π 2 2 i +1 j    cos 2  π 2 1 j  = cos  π − π 2 2 i +1 j  cos 2  π 2 1 j  = cos  π 2 2( j − i ) − 1 j  cos 2  π 2 1 j  ≤ cos  π 2 2 j  cos 2  π 2 1 j  = cos 2  π 2 1 j  − sin 2  π 2 1 j  cos 2  π 2 1 j  < 1 . 4. By item 1 of Lemma 4, if u ( i ) 1 < x ( j ) 2 , then i > 2 j − 2 3 . On the other hand, i + 1 < j , and due to these tw o inequalities, i + j ≥ 8 (21) holds. The seg ment  π ( j − 1) 2 j +1 , π i 2( i +1)  on the ϕ -a x is corr e s po nds to x ∈ [ u ( i ) 1 , x ( j ) 2 ]. Let ϕ ( ξ ) = π ( j − 1+ ξ ) 2 j +1 , where 0 ≤ ξ ≤ 3 i − 2 j + 2 2 i +2 . Then ϕ (0) = π ( j − 1) 2 j +1 and ϕ  3 i − 2 j + 2 2 i +2  = π i 2( i +1) . F or all x ∈ [ u ( i ) 1 , x ( j ) 2 ], the following inequality holds: | Z i ( x ) Z j ( x ) | ≤ | sin((2 j + 1) ϕ ( ξ )) | sin 2 ϕ ( ξ ) =   sin  (2 j + 1) π ( j − 1+ ξ ) 2 j +1    sin 2 π ( j − 1+ ξ ) 2 j +1 = | sin( π ( j − 1 + ξ )) | sin 2  π 2 − π ( 3 2 − ξ ) 2 j +1  = | sin( π ξ ) | cos 2 π ( 3 2 − ξ ) 2 j +1 . (22) Since i < j − 1 , w e get ξ ≤ 3 i − 2 j + 2 2 i +2 = 1 2  5 i +5 − 2( i + j ) − 3 2 i +2  = 1 2  5 − 2( i + j ) +3 i +1  ≤ 1 2  5 − 2( i + j ) +3 i + j 2  = i + j − 6 2( i + j ) < 1 2 and (22) results in | Z i ( x ) Z j ( x ) | ≤ | sin( π ξ ) | cos 2 π ( 3 2 − ξ ) 2 j +1 ≤   sin  π i + j − 6 2( i + j )    cos 2 3 2 π i + j <   cos  3 π i + j    cos 2 3 π 2( i + j ) =   cos 2 3 π 2( i + j ) − sin 2 3 π 2( i + j )   cos 2 3 π 2( i + j ) < 1 . The last inequality follows from (21). The lemma is proved. ⊓ ⊔ In all subsequent sta temen ts, the copies of mult iple ro ots of a p olynomia l are considered as distinct ro ots so that each p olynomia l of degree n has n distinct ro ots. Lemma 5. L et 0 < i < j − 1 . L et x 1 , x 2 , and x 3 b e t he thr e e sm al lest ro ots of t he p olynomial f ( x ) = Z i ( x ) Z j ( x ) and x 1 < x 2 ≤ x 3 . Then | f ( x ) | < 1 for al l x ∈ ]0 , x 3 ] . Pro of. Consider four cases. (a) x ( i ) 1 < u ( j ) 2 and u ( i ) 1 < x ( j ) 2 . This ca se is illustrated by Fig. 2. By item 3 of Lemma 4, | f ( x ) | < 1 holds for all x ∈ ]0 , u ( j ) 2 ]; by item 4, this inequality is also true on [ u ( i ) 1 , x ( j ) 2 ]. By item 2, u ( j ) 2 < u ( i ) 1 . On ] u ( j ) 2 , u ( i ) 1 [, we a ls o have | f ( x ) | < 1 , b ecause on this interv al | Z i ( x ) | < 1 and | Z j ( x ) | < 1. Th us, | f ( x ) | < 1 on ]0 , x ( j ) 2 ]. Since x ( j ) 2 = x 3 , the desired statement follows. (b) x ( i ) 1 < u ( j ) 2 and u ( i ) 1 ≥ x ( j ) 2 . In this ca s e, using item 3 of Lemma 4, we similarly obtain | f ( x ) | < 1 on ]0 , u ( i ) 1 ]. Since 0 < x 3 = x ( j ) 2 ≤ u ( i ) 1 , | f ( x ) | < 1 is true on ]0 , x 3 ] . (c) x ( i ) 1 ≥ u ( j ) 2 and u ( i ) 1 < x ( j ) 2 . By items 3 and 4 of Lemma 4, | f ( x ) | < 1 holds on ]0 , x ( i ) 1 ] and [ u ( i ) 1 , x ( j ) 2 ]. In addition, | f ( x ) | < 1 on ] x ( i ) 1 , u ( i ) 1 [, b ecause on this int er v al, | Z i ( x ) | < 1 and | Z j ( x ) | < 1 . Hence | f ( x ) | < 1 on ]0 , x ( j ) 2 ], where x ( j ) 2 = x 3 . (d) x ( i ) 1 ≥ u ( j ) 2 and u ( i ) 1 ≥ x ( j ) 2 . In this ca se, | f ( x ) | < 1 is g uaranteed on ]0 , x ( i ) 1 ]. If x ( i ) 1 ≥ x ( j ) 2 , then x 3 = x ( i ) 1 and the desired s tatement fo llows. In the opp osite case, x 3 = x ( j ) 2 and | f ( x ) | < 1 holds o n ] x ( i ) 1 , x ( j ) 2 ], beca use on this in terv a l | Z i ( x ) | < 1 and | Z j ( x ) | < 1. Therefore | f ( x ) | < 1 on ]0 , x 3 ], as needed. ⊓ ⊔ The next lemma provides a means of us ing Lemma 5 in the s ubs e quent pro ofs. Lemma 6. Supp ose that g ( x ) is a p olynomial with r e al c o efficients and x 1 ≤ x 2 ≤ x 3 ar e some of its distinct r e al r o ots. Supp ose that | g ( x ) | < 1 for al l x such that x 1 ≤ x ≤ x 3 . Then e ach of the p olynomials g ( x ) − 1 and g ( x ) + 1 has at le ast a p air of non-r e al r o ots. 9 Pro of. Under the assumptions o f Lemma 6, neither g ( x ) + 1 nor g ( x ) − 1 has an y real ro ot x ′ such that x 1 ≤ x ′ ≤ x 3 . On the other hand, the segment (po ssibly , degenerating int o a p oint) [ x 1 , x 3 ] contains at least tw o ro ots o f the deriv ative g ′ ( x ). Consequently , each o f the p olynomia ls g ( x ) + 1 and g ( x ) − 1 has no ro ot non-strictly b etw een t wo ro o ts of its deriv ative. Hence it has at least tw o non-real ro o ts. ⊓ ⊔ The following theorem determines which digra phs of type Γ ′′ n are ess ent ia lly cyclic. Its pro o f is based on Lemmas 5 and 6 and the subsequent L e mma 7. Theorem 3. L et L ′′ n b e t he L aplacian matrix of the digr aph Γ ′′ n whose ar cs form the H amiltonian cycle (1 , n ) , ( n, n − 1) , . . . , (2 , 1) , the p ath (1 , 2) , (2 , 3) , . . . , ( i − 1 , i ) , and the p ath ( i + 1 , i + 2) , . . . , ( n − 1 , n ) , wher e 1 ≤ i < n . Then : 1 . The char acteristic p olynomial of L ′′ n is ∆ L ′′ n ( λ ) = Z i ( λ ) Z n − i ( λ ) − ( − 1) n . 2 . If n is even , then Γ ′′ n is essen t ial ly cyclic for al l i ∈ { 1 , . . . , n − 1 } ex c ept for i = n 2 ; in the latter c ase the eigenvalues of L ′′ n ar e 4 cos 2 π k n and 4 cos 2 π k n +2 , k = 1 , . . . , n 2 . 3 . If n is o dd , then Γ ′′ n is ess en tial ly cyclic for al l i ∈ { 1 , . . . , n − 1 } exc ept for i = n − 1 2 and i = n +1 2 ; in the latter c ases t he eigenvalues of L ′′ n ar e 4 cos 2 π k n +1 , k = 1 , . . . , n. The only digraphs of type Γ ′′ n that ar e not e s sentially cyclic hav e their tw o vertices of indegree 1 (as well a s the tw o vertices of outdegree 1) at a maxim um p ossible distance. In other words, they ca n be obtained from Γ 2 n by the remov a l of tw o most distant a r cs fr om the same Hamiltonian cycle. These digraphs a re shown in Fig. 3. In Fig . 3(b), exactly one of the t wo dotted vectors must be an arc; the tw o resulting digraphs are obviously is omorphic. ✲ ✛ ✛ ✛ (a) ✲ q ❥ ② ❨ ❨ ☛ ✕ 1 2 n 2 n + 2 2 n n − 1 ✲ ✲ ✛ ✛ ✛ (b) ✲ q ❥ ❥ ❥ ② ❨ ❨ . . . . . . . . . . . ✒ . . . . . . . . . . . ■ ✕ 1 2 n − 1 2 n + 3 2 n n − 1 ✲ ❫ ✮ n + 1 2 Figure 3. Digraphs Γ ′′ n that are not essen tially cyclic: (a) n is even; i = n/ 2; (b) n is o dd; either i = ( n − 1) / 2 or i = ( n + 1) / 2, i. e., exactly on e of the t wo dotted vectors is an arc. Fig. 4 shows the shap e of the p oly nomials Z 2 i ( λ ) a nd Z i ( λ ) Z i +1 ( λ ), which differ by 1 from the characteristic p oly nomials ∆ L ′′ n ( λ ) of the digraphs Γ ′′ n that are not essentially cyclic due to Theorem 3. ✲ ✻ ✲ ✻ 1 1 − 1 − 1 0 0 (b) (a) Z i ( λ ) Z i +1 ( λ ) Z 2 i ( λ ) λ λ Figure 4. (a) The shap e of Z 2 i ( λ ); (b) t he shap e of Z i ( λ ) Z i +1 ( λ ). Pro of of Theorem 3. 1 . Expanding ∆ L ′′ n ( λ ) = det( λI − L ′′ n ) along the first row and using identit y (8 ) and the fact that for any sq uare matrices P and S , det P 0 R S ! = det P det S , one obtains 10 ∆ L ′′ n ( λ ) = ( λ − 2) Z i − 1 ( λ ) Z n − i ( λ ) − Z i − 2 ( λ ) Z n − i ( λ ) − ( − 1) n = Z i ( λ ) Z n − i ( λ ) − ( − 1 ) n , which proves item 1 of Theorem 3. T o prove items 2 and 3, w e need the following lemma. Lemma 7. 1. If i + j is even , then the e quation Z i ( x ) Z j ( x ) − 1 = 0 has only r e al r o ots if and only if i = j . In t he latter c ase , the r o ots ar e 4 cos 2 π k 2 j , 4 co s 2 π k 2 j +2 , k = 1 , . . . , j . 2. If i + j is o dd and i < j, then the e qu ation Z i ( x ) Z j ( x ) + 1 = 0 has only r e al r o ots if and only if i = j − 1 . In the latter c ase , Z i ( x ) Z j ( x ) + 1 = P 2 i + j  √ x  for al l x ≥ 0 (23) holds and t he r o ots ar e 4 co s 2 π k 2 j , k = 1 , . . . , i + j. Pro of of Lemma 7. W e fir s t prov e that under the requirements of Lemma 7, all the ro o ts are real. After that we show that otherwise there are at least tw o non-real ro ots. 1. If i = j , then Z i ( x ) Z j ( x ) − 1 = ( Z i ( x ) − 1 )( Z i ( x ) + 1 ), thus, Lemma 3 implies that all the ro ots are real, b elong to [0 , 4[, and ca n b e expressed as 4 cos 2 π k 2 j , 4 cos 2 π k 2 j +2 , k = 1 , . . . , j . This can a lso be obtained using Lemma 1, Eq. (5), and Catalan’s identit y for Chebyshev polyno mials (see [22, Eq . (1 . 1 ′ )]) ( P n ( x ) − 1)( P n ( x ) + 1) = P n − 1 ( x ) P n +1 ( x ). 2. Let j = i + 1. Then for x ∈ [0 , 4[ and ϕ = a rccos √ x 2 , using (10), (2), and (6), w e have Z i ( x ) Z j ( x ) + 1 = sin((2 i + 1) ϕ ) s in((2 i + 3) ϕ ) sin 2 ϕ + 1 = cos(2 ϕ ) − cos((4 i + 4) ϕ ) 2 sin 2 ϕ + 1 = sin 2 ((2 i + 2) ϕ ) sin 2 ϕ = P 2 i + j  √ x  = x j − 1 Y k =1  x − 4 cos 2 π k 2 j  2 = i + j Y k =1  x − 4 cos 2 π k 2 j  . This can also b e o btained using the identit y mentioned in the pro of of item 1. Thus, the ro ots of Z i ( x ) Z j ( x ) + 1 ar e 4 cos 2 π k 2 j , k = 1 , . . . , i + j ; they a re real and be long to [0 , 4[. Let us prov e that in the remaining cases, each of the equa tions under consideration has at least a pair of non-rea l ro ots. Let 0 < i < j − 1. By Lemma 5 , | Z i ( x ) Z j ( x ) | < 1 for all x ∈ ]0 , x 3 ], where x 1 , x 2 , a nd x 3 ( x 1 < x 2 ≤ x 3 ) are the three smallest r o ots of Z i ( x ) Z j ( x ). Hence by Lemma 6 each of the polyno mials Z i ( x ) Z j ( x ) + 1 and Z i ( x ) Z j ( x ) − 1 has at least a pair of non-real ro ots . The lemma is prov ed. ⊓ ⊔ In view of item 1 of Theo rem 3, items 2 and 3 follo w fr o m items 1 and 2 of Lemma 7, resp ectively . ⊓ ⊔ 4.4. The digraphs Γ n with m ( n < m < 2 n − 2) arcs Let us summarize the ab ov e results. Accor ding to Theorems 1 to 3: Γ 1 n is es sentially cyclic; the digraphs Γ ′′ n are ess ent ia lly cy c lic except for the cases sp ecified in Theorem 3; Γ ′ n and Γ 2 n are not esse n tially cyclic . The following theor em answers the question of essential cyclicit y for the remaining digraphs with r ing structure, in which the Hamiltonian coun ter- cycle la cks mor e than t wo ar cs. According to this theorem, all such digr aphs are essentially cyclic. Theorem 4. L et Γ n b e a digr aph on n > 3 vertic es c onstitut e d by the Hamiltonian cycle { (1 , n ) , ( n, n − 1) , . . . , (2 , 1) } and the opp osite cycle { (1 , 2) , (2 , 3) , . . . , ( n − 1 , n ) , ( n, 1) } in which i (2 < i < n ) arbitr ary ar cs ar e missing. Then : 1. The char acteristic p olynomial of the L aplacia n matrix L n of Γ n is 11 ∆ L n ( λ ) = K Y k =1 Z i k ( λ ) − ( − 1) n , (24) wher e i 1 , . . . , i K ar e the p ath lengths in the de c omp osition of the cycle { (1 , n ) , ( n, n − 1) , . . . , (2 , 1) } into the p aths linking t he c onse cutive vert ic es of inde gr e e 1 in Γ n . 2. Γ n is ess en tial ly cyclic. Pro of. 1. The pro o f is quite similar to that of item 1 of Theorem 3. 2. W e need the following lemma, which extends Lemma 5. Lemma 8. L et f ( x ) = Q K k =1 Z i k ( x ) , wher e K > 2 and i k > 0 , k = 1 , . . . , K . 1 . L et x 1 , x 2 , and x 3 b e t he smal lest 3 r o ots of f ( x ) and x 1 ≤ x 2 ≤ x 3 . Then | f ( x ) | < 1 for al l x ∈ ]0 , x 3 ] . 2 . Each of the p olynomials f ( x ) − 1 and f ( x ) + 1 has at le ast a p air of non-r e al r o ots. Pro of of Lemma 8. 1 . The pr o of pro ceeds b y induction o n K . W e first conside r the step of inductio n and then come bac k to its base. Assume that the requir ed statemen t is true for all K < K 0 . Let us prov e that it is a lso true for K = K 0 . Consider any pro duct o f the form f ( x ) = Q K 0 k =1 Z i k ( x ) . Without loss of generality , assume that i K 0 = min ( i 1 , . . . , i K 0 ) . (25) Then f ( x ) = Z i K 0 ( x ) f 0 ( x ) , (26) where f 0 ( x ) = Q K 0 − 1 k =1 Z i k ( x ). Let x 0 1 , x 0 2 , and x 0 3 be the three s mallest ro o ts of f 0 ( x ) and x 0 1 ≤ x 0 2 ≤ x 0 3 . By the assumption, 4 | f 0 ( x ) | ≤ 1 on ]0 , x 0 3 ] . (27) By (16), the s mallest r o ot of Z i ( x ) decreases w ith the increase of i . Therefore (25) implies tha t x 1 = x 0 1 , x 2 = x 0 2 , and x 3 = min( x 0 3 , x ( K 0 ) ), where x ( K 0 ) is the smallest r o ot of Z i K 0 ( x ). Having in mind (27) and that | Z i K 0 ( x ) | < 1 on ]0 , x ( K 0 ) ] (which follows from Lemma 1 and (2)), we hav e that | f 0 ( x ) | ≤ 1 and | Z i K 0 ( x ) | < 1 on ]0 , x 3 ]. Hence, by (26), | f ( x ) | < 1 on ]0 , x 3 ], thus, the inductio n step is complete. W e now tur n to the base of induction. Let K = 3 and f ( x ) = Q 3 k =1 Z i k ( x ) . Without los s of generality , assume that i 3 ≤ i 2 ≤ i 1 . Then f ( x ) = Z i 3 ( x ) f 0 ( x ) , wher e f 0 ( x ) = Z i 2 ( x ) Z i 1 ( x ). Let x 0 1 , x 0 2 , and x 0 3 be the three smallest ro o ts o f f 0 ( x ) ordered as follows: x 0 1 ≤ x 0 2 ≤ x 0 3 . Conside r three ca ses. (a) i 2 < i 1 − 1. Then, by Lemma 5, | f 0 ( x ) | < 1 for all x ∈ ]0 , x 0 3 ]. Now, applying the ab ove induction step, one has | f ( x ) | < 1 for all x ∈ ]0 , x 3 ], as needed. (b) i 2 = i 1 − 1. In this case, by (2 3), f 0 ( x ) = Z i 2 ( x ) Z i 1 ( x ) = P 2 i 2 + i 1  √ x  − 1 (see also Fig . 4(b)) and we only hav e | f 0 ( x ) | ≤ 1 on ]0 , x 0 3 ]. How ever, the ca se of the weak inequality | f 0 ( x ) | ≤ 1 is covered by the ab ov e induction step (see (27)), thereby , this step provides | f ( x ) | < 1 on ]0 , x 3 ]. (c) i 2 = i 1 . In this case, b y Lemma 1, f 0 ( x ) = Z 2 i 2 ( x ) = P 2 2 i 2 ( √ x ), and | f 0 ( x ) | > 1 is p ossible for so me x ∈ ]0 , x 0 3 ] (cf. Fig. 4(a )). Let x ( i 3 ) be the smallest ro ot o f Z i 3 ( x ). Then by (1) for every integer k > 0 and x ∈ ]0 , 1] we have P 2 2 k ( √ x ) = sin 2  (2 k + 1) a rccos √ x 2  1 − x 4 ≤ 1 1 − x 4 = 1 + x 4 − x ≤ 1 + x 3 . (28) Moreov er , since x ( i 3 ) ≤ 1, 3 Here, as earl ier, w e distinguish the copies of ev ery multiple ro ot of a polynomial. 4 The inequalit y | f 0 ( x ) | < 1 i s giv en here in a w eakene d form for the subsequen t use of the induction step in the case where the strict inequality is not satisfied. 12 | Z i 3 ( x ) | ≤ | Z 1 ( x ) | = 1 − x for all x ∈ ]0 , x ( i 3 ) ] . (29) Indeed, | Z i 3 (0) | = | Z 1 (0) | = 1; by (9), | Z i 3 ( x ) | ′ x =0 = − n ( n +1) 2 ≤ − 1 = | Z 1 ( x ) | ′ x =0 . Now the assumption that | Z i 3 ( x ) | > | Z 1 ( x ) | at some x ∈ ]0 , x ( i 3 ) ] implies that | Z i 3 ( x ) | has an inflection on ]0 , x ( i 3 ) [, which is impo ssible b ecause Z i 3 ( x ) has i 3 real ro ots and x ( i 3 ) is the smallest one. Using (28) and (29 ) for every x ∈ ]0 , x ( i 3 ) ] we hav e | f ( x ) | = | Z i 3 ( x ) P 2 2 i 2 ( √ x ) | < (1 − x )(1 + x ) < 1 . Finally , i 3 ≤ i 2 = i 1 implies that x 3 = x ( i 3 ) , thereb y | f ( x ) | < 1 for a ll x ∈ ]0 , x 3 ]. 2. Item 2 follows from item 1 a nd Lemma 6. Le mma 8 is prov ed. ⊓ ⊔ Now item 2 of Theorem 4 follows fr om (24) and item 2 of Lemma 8. ⊓ ⊔ Corollary of Theorem 4. If for t wo digr aphs of the t yp e describ e d in The or em 4 , the p ath lengths i 1 , . . . , i K in t he de c omp osition of { (1 , n ) , ( n, n − 1) , . . . , (2 , 1) } into the p aths linking the c onse cutive vertic es of inde gr e e 1 differ only by the or der of the c orr esp onding p aths in the de c omp osition , then these digr aphs have t he same L aplacian sp e ctrum. Thu s, the answer to the question in the title of this pa per is as follows. The digraphs Γ n with r ing structure, which co nsist of t wo o ppo site Hamiltonian cycles fr o m one of whic h some arcs can be removed, are essentially cyclic, except for (up to isomorphis m) thr ee digraphs: • Γ 2 n , where no arcs are remov ed (Fig. 1(b)); • Γ ′ n , where one arc is remov ed (Fig. 1(c)); • Γ ′′ n with t wo most distant 5 arcs removed (Fig. 3). A by-product of this work is the following theorem on the Cheb yshev p olynomia ls of the second kind. Theorem 5 . L et h ( x ) = Q K k =1 P 2 i k ( x ) + ( − 1) p , wher e p ∈ { 0 , 1 } , P 2 i k ( x ) ar e the Chebyshev p olynomials of the se c ond kind s c ale d on ] − 2 , 2[ ( se e (1)) , K ≥ 1 , and i k > 0 , k = 1 , . . . , K . Then h ( x ) has only r e al r o ots if and only if (a) K = 1; t he r o ots ar e  ± 2 c o s π k 2 j +1+( − 1) k + p   k = 1 , . . . , j  , wher e j = i 1 or (b) K = 2 , i 1 = i 2 , and p = 1; the r o ots ar e  ± 2 c o s π k 2 j ; ± 2 cos π k 2 j +2   k = 1 , . . . , j  , wher e j = i 1 or (c) K = 2 , | i 1 − i 2 | = 1 , and p = 0; the r o ots ar e 6  ± 2 c o s π k 2 j   k = 1 , . . . , 2 j − 1  , whe r e j = max( i 1 , i 2 ) . Pro of. By Lemma 1, h ( x ) = Q K k =1 Z i k ( x 2 ) + ( − 1 ) p . In the case (a), h ( x ) ha s o nly rea l ro ots by virtue of Lemma 3. Suppo se that K = 2. If | i 1 − i 2 | > 1, then Lemmas 5 and 6 imply that h ( x ) has at least a pair of non-real ro o ts. If i 1 = i 2 and p = 1 (the case (b)), then by item 1 of Lemma 7, h ( x ) has only real ro ots. If i 1 = i 2 and p = 0, then h ( x ) = Z 2 i 1 ( x 2 ) + 1 has no rea l ro o ts. If | i 1 − i 2 | = 1 and p = 0 (the ca se (c)), then b y item 2 of Lemma 7, h ( x ) has only real ro o ts. If | i 1 − i 2 | = 1 and p = 1, then using the notatio n i = min( i 1 , i 2 ), by item 2 of Lemma 7 we hav e h ( x ) = Z i ( x 2 ) Z i +1 ( x 2 ) − 1 = P 2 2 i +1 ( x ) − 2 . Since P 2 2 i +1 ( x ) has a maximum on ]0 , 1[ and on this interv al, 5 F or the exact meaning of “most dis tan t,” see Theorem 3. 6 In this expression, each element app ears t wice, which corresp onds to multiplicit y 2 of every ro ot of h ( x ) in the case (c) . 13 P 2 2 i +1 ( x ) = sin 2  (2 i + 2) ar ccos x 2  1 − x 2 4 < 1 1 − 1 4 = 4 3 , h ( x ) has a negative maximum, consequently , it has at least a pair of non-real ro ots. Finally , if K > 2, then b y item 2 of Lemma 8, h ( x ) has a t least t wo non-real ro ots. The express ion for the ro o ts in the ca s e (a) is pr ovided by Lemma 3; in the cases (b) and (c) they are easily obtained using the identit y P n − 1 ( x ) P n +1 ( x ) + 1 = P 2 n ( x ) (see [22, Eq. (1 . 1 ′ )]) or Lemma 7. ⊓ ⊔ 5. On the essen tial cyc licit y of w eigh ted digraphs In this section, we study the essential cyclic ity of s imple weigh ted digraphs with ring structure. Recall that the Laplacian matrix of a w eig h ted digraph Γ with strictly po sitive arc weigh ts is the matrix L = L (Γ) = ( ℓ ij ) in which, for j 6 = i , ℓ ij equals minus the weigh t of ar c ( i, j ) in Γ and ℓ ij = 0 if Γ has no ( i, j ) arc, the diagonal en tries of L (Γ) b eing such that the row sums are zero. Some sp ectra l prop erties of the Laplacian matrices of weigh ted digra phs were studied in [1–3, 6 , 8 ], paper s cited therein, and, with a different definition of the Laplacian matrix, in [13]. First, consider a n arbitra ry w eighted directed cycle C 3 on three vertices, se e Fig. 5(a). Recall that by Theorem 1 the unweighte d directed cycle is ess e ntially cyclic. ✐ ❯ ✕ ✐ ❯ ✕ ⑦ ❂ ✻ 1 1 2 2 3 3 a a b b c c α γ β ( a ) ( b ) Figure 5. W eighted digraphs on three vertices: (a) a w eighted cycle C 3 ; (b) a complete digraph K 3 . Prop ositio n 1. The weighte d 3 -cycle C 3 is essential ly cyclic if and only if the squar e r o ots √ a, √ b, and √ c of its ar c weights satisfy the strict triangle ine quality , namely : √ a < √ b + √ c, √ b < √ a + √ c, and √ c < √ a + √ b. Prop ositio n 1 follows fro m Theo r em 6 b elow. It can b e in terpre ted as follows: in an es sentially cyclic digraph C 3 , the weigh t of a ny arc is not large eno ugh to “ ov erp ow er” the remaining par t of the cycle. Or, in more precise terms, C 3 is essentially cyclic whenever the squa re r o ots of its a r c w eights ar e the lengths of the sides of a non-deg e ne r ate triangle. Now consider the complete weight ed digra ph (without lo ops) on three vertices, K 3 (Fig. 5(b)). The Laplacian characteristic equatio n of this digraph, det     λ − b − γ b γ α λ − c − α c a β λ − a − β     = 0 , reduces to 14 λ ( λ 2 − ( a + b + c + α + β + γ ) λ + ( ab + bc + ca + αβ + β γ + γ α + aα + bβ + cγ )) = 0 . The digraph is essentially cyclic if and only if D < 0 , where D = ( a + b + c + α + β + γ ) 2 − 4( ab + b c + ca + αβ + β γ + γ α + aα + b β + cγ ) . Equiv a lent ly , D = ( a − α ) 2 − 2( a − α )( b + c − β − γ ) + ( b − c − β + γ ) 2 . This quadratic trinomial in a − α is negative iff its ro ots ar e r e a l and a − α lies strictly b etw een them. The ro ots ( a − α ) 1 , 2 = ( b + c − β − γ ) ± p ( b + c − β − γ ) 2 − ( b − c − β + γ ) 2 = ( b − β ) + ( c − γ ) ± 2 p ( b − β )( c − γ ) (30) are real and unequal iff ( b − β )( c − γ ) > 0 . (31) Assuming that (31) is satis fie d, first consider the ca se of b > β and c > γ . In this case, (3 0) re duces to the en tirely r e al expression ( a − α ) 1 , 2 = ( p b − β ± √ c − γ ) 2 . Then the inequality D < 0, i. e. the ess e n tial cyclicity of K 3 , amounts to    p b − β − √ c − γ    < √ a − α < p b − β + √ c − γ , which is the stric t triang le inequality for √ a − α , √ b − β , and √ c − γ . The case of b < β and c < γ is consider ed s imila rly; as a result we o btain the following theorem. Theorem 6. L et the matrix of ar c weights of a weighte d digr aph Γ b e W =     0 b γ α 0 c a β 0     . Then Γ is essential ly cyclic if and only if either (i) √ a − α, √ b − β , and √ c − γ ar e r e al and satisfy the strict triangle ine quality or (ii) √ α − a, √ β − b, and √ γ − c ar e r e al and satisfy t he st rict triangle ine quality. This criterion corr esp onds to a cer tain in tuitive sense of essen tial cyclicity . According to Theorem 6, for a complete weighted digra ph on thre e vertices, K 3 , to b e ess ent ially cyclic, it is ne c e ssary that it o bey s the strict triang le inequality applied to certain v alues rather a ttached to the vertices than to the pairs o f them. Indee d, suc h a v alue is the squa re ro o t of the weight difference for the tw o a rcs con verging to the same vertex. The pro blem of c har acterizing the essentially cyclic weight ed digr aphs b ecomes m uch mor e difficult with the increase of the num b er of vertices. How ever, some of the corr e sp o nding conditions inv olve the triangle inequality fo r the r o ots of the arc weigh ts as w ell. Consider the fir st w eighted dig r aph in Fig. 6. Note that the cor resp onding unw eighted dig raph is essentially cy clic b y Theorem 4. The Laplacian characteristic e q uation for the weigh ted digraph, det       λ − p − 1 1 0 p 0 λ − y y 0 0 0 λ − 1 1 1 0 0 λ − 1       = 0 , as well as that for the seco nd w eighted dig raph shown in Fig. 6, reduce s to the for m 15 ❄ ✻ ✻ ❄ 1 1 1 1 p y 2 3 4 1 y 1 p 1 1 2 3 4 ✲ ❄ ✛ ✲ ❄ ✛ Figure 6. Tw o cospectral w eighted digraphs with ring stru ct ure on four vertices. λ ( λ 3 − ( y + q ) λ 2 + ( q y + q ) λ − ( q y + 1 )) = 0 , where q = p + 3 . The digraph is essentially cyclic whenever D < 0 , where D = 4( b 2 − 3 c ) 3 − (2 b 3 − 9 bc + 2 7 d ) 27 , (32) b = − ( y + q ) , c = q y + q , and d = − ( q y + 1 ) . Substituting these express ions for b , c , and d in (32) we obtain D = q ( q − 4) y 4 − (2 q 3 − 8 q 2 + 4) y 3 + q ( q 3 − 2 q 2 − 8 q + 6) y 2 − 2 q ( q + 2 )( q − 3) 2 y + ( q + 1)( q − 3) 3 . (3 3) T o solve the inequality D < 0 w.r .t. y , one can find the real ro o ts of the p oly nomial (3 3) trea ting q as a parameter. How ever, the expressions of these ro o ts as functions in q a re ra ther cumber some, as well as the inv erse representations of q via y . F or exa mple, when p = 3, the ro ots of the equation D = 0 are: y 1 , 2 =  37 − Q ± p 290 − 36 z − 5 04 /z + 34 54 /Q  / 12 , where Q = p 36 z + 1 45 + 504 /z, z = 0 . 5 3 q 671 + 65 √ 65 . Therefore it do es not seem to be ea sy to formulate a simple criterion (suc h as Theorem 6) of essential cyclicity for the digraphs of this kind. Solving the pr oblem n umerica lly , we obtain that for p = 3, the digraph is essentially cyclic, i. e. D < 0, when 0 . 266 < y < 2 . 441 (approximately). So the essential cyclicity can b e suppres sed b y either incr ease or decrease of the arc weigh t y . Finally , consider a weighted cycle on four vertices w ith t wo v ariable weight s (Fig. 7). ❄ ✻ ✲ ✛ 4 9 a x 1 2 3 4 Figure 7. A cycle on four vertices with tw o va riable arc w eights, a and x . The corresp onding Laplacian characteristic p o ly nomial is f ( λ ) = λ ( λ 3 − (13 + x + a ) λ 2 + (36 + 1 3 x + 1 3 a + ax ) λ − 3 6 x − 36 a − 13 ax ) . The b oundary of the domain on the ( a, x ) plane cor resp onding to the ess e n tially cyclic digr aphs is sp ecified by the equa tio n 16 − a 2 x 2 ( x − a ) 2 + 26( x + a )  ax ( x − a ) 2 + 25( x 2 + a 2 ) + 58 ax + 900  + 870 a 2 x 2 − 241( x 2 + a 2 )(2 ax + 25 ) − 25( x 4 + a 4 ) − 39 34 ax − 3 2 400 = 0 . The polyno mial on the left-hand side is not the pr o duct o f po lynomials with ratio na l co efficients a nd Figure 8. F or the weigh ted d igraph sh o wn in Fig. 7, the domain where the sq uare roots of the three smaller arc w eights satisfy th e t riangle inequ alit y is filled in d ark grey; the domain corresp onding to the essen tially cyclic digraphs is the union of the ab ove “tria ngle inequalit y domain” and the four fringing d omains filled in ligh t grey . smaller degrees . The solutions ( a, x ) of the ab ov e equation in the non- negative quadrant are plotted in Fig. 8 with the “ro o ted” s cales √ a and √ x . The domain cor resp onding to the essentially cyc lic digr aphs is filled. The sub domain filled in dark grey is the lo cus o f p oints ( √ a, √ x ) for which the sq uare ro ots of the three sma llest ar c weights satisfy the tria ng le inequality . The lo cus of p oints ( √ a, √ x ) that corresp ond to the essentially cyclic digr aphs is wider: it also cont ains the four sub doma ins filled in light grey . Thu s, for this cyclic digraph, the tria ngle inequality for the square r o ots of the three smaller arc weigh ts is a sufficient, but not nece s sary co ndition of e s sential cyclicity . In other words, the r equired criterion of essential cyclicity is a kind of re laxe d triangle ineq uality . This relaxed inequality turns into the triangle ine q uality as the fourth (larg est) ar c weigh t tends to infinity o r the smalles t weigh t tends to zero; the relaxation is maxima l when the la rgest weigh t b eco mes equal to the second la rgest weigh t. It can b e c o njectured that the triang le inequality for the sq uare ro ots of the three smallest ar c weighs is a sufficient co nditio n of essential cyclicity for the whole class of weigh ted 4-cycles. As one can see, even for the w eig h ted digraphs on four vertices, the problem of characterizing essential cyclicity in terms of g raph topolo gy is non-trivial. 17 6. Concluding remarks W e conc lude with tw o side re marks. 1. Acco rding to the matrix tree theorem (see , e.g., Theorem 16.9 in [17 ]), for e very digraph Γ, the cofactor of each ent r y in the i th row of the Lapla cian matr ix is eq ua l to the num b er of spanning conv er ging trees (also called in-arb ore s cences) ro o ted at vertex i . The total num b er of in-arb or escences is equal to the sum of the cofactors in any c olumn of L . Thu s, the matrix tree theor em provides a general approa ch to computing the num b er of in- arb ores c e nces in a digraph and the num b er of spanning tr ees in a graph. F or certain cla sses of graphs , this approach lea ds to explicit for mulas which can be obtained using the Chebyshev p olynomials of the second kind. In [5], this metho d w as applied to the wheels, fans, M¨ obius ladders, etc. In [25 ], the Chebyshev po lynomials were used for finding the num b e r o f spanning trees for certain classes of graphs including circulant graphs with fixed and non-fixed jumps. The re s ults of the prese n t pap er can be us ed to o btain representations for the n umber of co nv erging tr ees in the digraphs with ring structure. Suppo se that t n i is the n umber of spanning con verging tr e e s in the digraph who se Laplacia n matrix is L ′′ n (Eq. (15)). Then summing up the cofactors of the la st column o f L ′′ n and having in mind that (i) det M n = ( − 1) n Z n (0) = ( − 1) n P 2 n (0) = 1, (ii) P n ( x ) = det( xI − C n ), where C n =              0 1 0 1 0 1 . . . . . . . . . . . . . . . . . . 1 0 1 0 1 0              (see, e. g., [2 0, Theor e m 1.11]), a nd (iii) P k (2) = k + 1 (whic h equals the limit from the left at x = 2 of the expressio n (2)) we obtain the following repres ent a tio n for t n i : t n i = i − 1 X k =0 P k (2) + n − i − 1 X k =0 P k (2) = 1 2  i 2 + n + ( n − i ) 2  . (34) Since t n i is the pro duct of the eig env a lues of L ′′ n , ex cept for a zero eigenv alue , for the non-tr ivial digraphs with ring s tr ucture that hav e completely real sp ectra, Eq . (34) and Theor em 3 pro vide the following expr essions fo r t n i and simultaneously trig onometric iden tities: t n n/ 2 =   n/ 2 − 1 Y k =1 2 co s π k n n/ 2 Y k =1 2 co s π k n + 2   2 = n ( n + 2) 4 , if n is even ; t n ( n − 1) / 2 = t n n +1 / 2 =   ( n − 1) / 2 Y k =1 2 co s π k n + 1   4 = ( n + 1) 2 4 , if n is o dd . 2. Let L c n be the Laplacian matrix of the undirected cy cle on n vertices. By suitable index ing of the vertices, L c n can be presented as a matrix different from M n (7) in the (1 , 1) entry , which in L c n is 1. Expanding det ( λI − L c n ) a long the first row and using (8), Lemma 1, and (6), for all λ ∈ ]0 , 4] one has: ∆ L c n ( λ ) = ( λ − 1) Z n − 1 ( λ ) Z n − 2 ( λ ) = Z n ( λ ) + Z n − 1 ( λ ) = P 2 n ( √ λ ) + P 2( n − 1) ( √ λ ) = √ λ P 2 n − 1 ( √ λ ) = λ n − 1 Y k =1  λ − 4 cos 2 π k 2 n  = n Y k =1  λ − 4 cos 2 π k 2 n  . 18 Thu s, the ro ots of ∆ L c n are (4 cos 2 π k 2 n , k = 1 , . . . , n ). O bviously , (4 sin 2 π k 2 n , k = 0 , . . . , n − 1) is a different representation of the same sp ectr um. The latter representation was taken as “r eady-made” and then prov ed in [15] and [4]. The a bove r eduction to Cheb yshev p o lynomials provides a deriv ation of this r esult. Another deriv a tion can b e obtained using item 1 of Theorem 1 and the representation C = ~ C ∪ ~ C n − 1 , where ~ C is the dir ected cycle and ~ C n − 1 its ( n − 1)st p ow er. On connections of the adjacency characteristic p olyno mia ls with Cheb ys hev p olyno mials, see [14, § 2.6 ]. Ac kno wledgemen ts This work was partially supp orted b y RFBR Grant 09-07- 00371 and the RAS P rogr am “Dev elopment of Net work and Logical Control in Conflict and Co op er a tive Environmen ts.” References [1] R.P . Agaev, P .Y u. Cheb otarev, The matrix of maximum ou t fo rests of a digraph an d its applicatio ns, Autom. Remote Control 61 ( 2000) 1424–1450. [2] R.P . Agaev, P .Y u. 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