Faber-Krahn inequalities for first Dirichlet eigenvalues of combinatorial $p$-Laplacian on graphs with boundary

F ABER-KRAHN INEQUALITIES F OR FIRST DIRICHLET EIGENV ALUES OF COMBINA TORIAL p -LAPLA CIAN ON GRAPHS WITH BOUND AR Y W ANKAI HE AND CHENGJIE YU 1 Abstract. In this pap er, we obtain sharp F ab er-Krahn inequalities for the first Dirichlet eigen v alue of the com binatorial p -Laplacian on connected graphs with a fixed n um b er of v ertices or with a fixed n umber of edges. More precisely , w e sho w that the minimum of the first p -Diric hlet eigen v alues of connected graphs with b oundary that consist of n v ertices or n edges is achiev ed only on the tadpole graph T n, 3 when p > 1. 1. Intr oduction In spectral geometry , the classical F ab er-Krahn inequalit y (see [4, P . 87]) says that the minim um of the first Dirichlet eigenv alues for smooth bounded Euclidean domains with a fixed volu me is only ac hiev ed on the round ball with the same v olume. Exp erts also tried to extend this imp ortan t inequality to discrete setting whic h mak es it an imp ortan t theme in extremal sp ectral graph theory . In [6], F riedman considered F ab er-Krahn inequalit y for regular trees. He for- m ulated the problem of finding the F ab er-Krahn inequalit y for domains in a homogeneous tree with a fixed total length and conjectured that the minim um is ac hieved by a geo desic ball in the homogeneous tree. In [18], Pruss disprov ed F riedman’s conjecture. In a series of w orks [14, 15], Leydold completely solved F riedman’s problem. In [7], F riedman established F ab er-Krahn inequalities for eigen v alues of the combinatorial Laplacian on graphs with a fixed n um b er of ver- tices. The result was recen tly extended by the second named author and Yingtao Y u [21] to Steklo v eigen v alues. In [13], Katsuda and Urak a w a obtained a sharp F ab er-Krahn inequalit y for the first Dirichlet eigenv alue of the normalized combinatorial Laplacian. In [2], Bıyık o˘ glu and Leydold formulated the general problem of finding graphs satisfying the so called F ab er-Krahn prop erty in a certain class of graphs and solv ed the problem for several classes of graphs such as trees with fixed n um b ers of v ertices 2020 Mathematics Subje ct Classific ation. Primary 05C35; Secondary 35R02. Key wor ds and phr ases. F ab er-Krahn inequalit y , Dirichlet eigenv alue, combinatorial p - Laplacian. 1 Researc h partially supported by GDNSF with con tract no. 2025A1515011144 and 2026A1515012267. 1 2 He & Y u and b oundary v ertices, trees with fixed num b ers of v ertices and b oundary v ertices and with a fixed lo w er b ound on degree of in terior v ertices, and trees with a given degree sequence. In [22, 24, 23, 20, 17], the authors further considered the problem form ulated in [2] for other classes of graphs. In [5] and [19], the authors related the F ab er-Krahn inequality on graphs to heat k ernel b ounds. In [16], the authors obtained interesting low er b ounds for the first Diric hlet eigen v alue of a graph in terms of its diameter. In [1] and [10], the authors considered related estimates on Diric hlet eigen v alues for subgraphs of the in teger lattice graph. In this pap er, motiv ated by the w ork of Katsuda-Urak aw a [13], w e consider F ab er-Krahn inequalities for the un-normalized com binatorial Laplacian on con- nected graphs with a fixed ”v olume”. Let’s first recall the main result in [13]. Theorem 1.1 (Katsuda-Urak aw a [13]) . L et G b e a c onne cte d gr aph with b ound- ary. Supp ose that | E ( G ) | = n ≥ 4 . Then, λ nor 1 ( G ) ≥ λ nor 1 ( T n, 3 ) and the e quality holds if and only if G = T n, 3 . Here a connected graph is considered as a graph with b oundary b y taking its p endan t v ertices, i.e. vertices of degree one, as the b oundary v ertices, and T n, 3 is a tadpole graph on n vertices with the head of size 3. F or the definition of tadp ole graph, see Section 2. Moreo v er, λ nor 1 ( G ) means the first Diric hlet eigenv alue for the normalized com binatorial Laplacian of G which is defined as ∆ nor f ( x ) = 1 deg( x ) X y ∼ x ( f ( x ) − f ( y )) . In this pap er, we consider F ab er-Krahn inequalit y on graphs with b oundary for the un-normalized com binatorial p -Laplacian: ∆ p f ( x ) = X y ∼ x | f ( x ) − f ( y ) | p − 2 ( f ( x ) − f ( y )) for p > 1. More precisely , w e obtain the following t w o sharp F ab er-Krahn inequal- ities for the first Diric hlet eigen v alue λ 1 ,p of ∆ p on graphs with a fixed num b er of v ertices or with a fixed n um b er of edges. Theorem 1.2. L et G b e a c onne cte d gr aph with b oundary on n ≥ 4 vertic es and p > 1 . Then, λ 1 ,p ( G ) ≥ λ 1 ,p ( T n, 3 ) and the e quality holds if and only if G = T n, 3 . Theorem 1.3. L et G b e a c onne cte d gr aph with b oundary that c onsists of n ≥ 4 e dges and p > 1 . Then, λ 1 ,p ( G ) ≥ λ 1 ,p ( T n, 3 ) F ab er-Krahn inequality 3 and the e quality holds if and only if G = T n, 3 . Note that λ 1 , 1 ( G ) = lim p → 1 + λ 1 ,p ( G ) . (see [8]) So, the F ab er-Krahn inequalities in Theorem 1.2 and Theorem 1.3 can b e ex- tended to λ 1 , 1 ( G ). Ho w ever, the rigidity part do es not hold in the case p = 1. Indeed, by that λ 1 , 1 ( G ) = h D ( G ) where h D ( G ) is Diric hlet Cheeger constant of G (see (2.2)), it is clear that λ 1 , 1 ( G ) ≥ 1 n − 1 = λ 1 , 1 ( T n, 3 ) . When G is a connected graph with boundary on n ≥ 4 vertices, the equalit y holds if and only if G has only one p endant vertex. When G consists of n edges, w e can characterize the rigidit y more explicitly . Theorem 1.4. L et G b e a c onne cte d gr aph c onsisting of n ≥ 4 e dges. Then, λ 1 , 1 ( G ) ≥ 1 n − 1 with e quality if and only if G = T n,i for some i with 3 ≤ i < n . Our argumen ts to pro v e Theorem 1.2 and Theorem 1.3 are similar to that of Katsuda-Urak aw a [13] by taking surgeries on graphs decreasing the Rayleigh quotien t. Ho w ever, the surgeries we tak e are muc h simpler than that of Katsuda- Urak aw a [13]. In [12], we ga ve a muc h simpler pro of of Katsuda-Urak aw a’s result (Theorem 1.1) and extended it to normalized combinatorial p -Laplacian by using the same surgery . The organization of the rest of paper is as follows: In Section 2, we in troduce some basic definitions and notations and obtain some simple sp ectral prop erties for the first Diric hlet eigen v alue of tadp ole graphs; In Section 3, w e prov e Theorem 1.2 and Theorem 1.3. 2. Preliminaries In this section, w e in tro duce some basic definitions and notations, and obtain some simple sp ectral prop erties for the first Dirichlet eigen v alue of tadp ole graphs. Let G b e a nontrivial connected finite graph and B ( G ) := { x ∈ V ( G ) | deg( x ) = 1 } whic h is viewed as the b oundary of G . The set Ω( G ) := V ( G ) \ B ( G ) 4 He & Y u is viewed as the in terior of G . F or p > 1, the un-normalized com binatorial p - Laplacian on G is defined as ∆ p,G f ( x ) = X y ∼ x | f ( x ) − f ( y ) | p − 2 ( f ( x ) − f ( y )) , ∀ x ∈ V ( G ) , where f ∈ R V ( G ) . The un-normalized combinatorial p -Laplacian is also called the com binatorial p -Laplacian for simplicit y . A real n um b er λ is called a p -Dirichlet eigen v alue of G if the following Diric hlet b oundary v alue problem:  ∆ p,G f ( x ) = λ | f | p − 2 f ( x ) x ∈ Ω( G ) f ( x ) = 0 x ∈ B ( G ) has a nonzero solution f . In this case, f is called a p -Diric hlet eigenfunction of G . W e denote the smallest p -Dirichlet eigenv alue of G as λ 1 ,p ( G ) and call it the first p -Diric hlet eigenv alue of G . An eigenfunction of λ 1 ,p ( G ) is called a first p -Dirichlet eigenfunction of G . The first p -Diric hlet eigenv alue of G can b e c haracterized b y the minimum of the p -Rayleigh quotient: (2.1) λ 1 ,p ( G ) = min f ∈ C B ( G ) \{ 0 } R p,G [ f ] and the minim um is only ac hiev ed b y first p -Diric hlet eigenfunctions. Here C B ( G ) = n f ∈ R V ( G )    f | B ≡ 0 o , and R p,G [ f ] = ∥ d f ∥ p p,G ∥ f ∥ p p,G with ∥ d f ∥ p p,G = X { x,y }∈ E ( G ) | f ( x ) − f ( y ) | p and ∥ f ∥ p p,G = X x ∈ V ( G ) | f | p ( x ) = X x ∈ Ω( G ) | f | p ( x ) since f ∈ C B ( G ) . The definition of the p -Laplacian is subtle for p = 1 (see [4] for example). How- ev er, the first 1-Dirichlet eigen v alue λ 1 , 1 can b e also characterized by minimum of the Ra yleigh quotient (2.1) for p = 1. It is also w ell-kno wn (see [9, 11] for example) that λ 1 , 1 ( G ) = h D ( G ) where (2.2) h D ( G ) = inf ∅ = U ⊂ Ω( G ) | E ( U, U c ) | | U | F ab er-Krahn inequality 5 is the Dirichlet Cheeger constan t of G . F or an y nonempty set U ⊂ Ω( G ), setting f = 1 U (the characteristic function on U ) in (2.1), one obtains λ 1 ,p ( G ) ≤ h D ( G ) directly . If B ( G )  = ∅ and p > 1, then the first p -Diric hlet eigenv alue λ 1 ( G ) is p ositive and the first p -Diric hlet eigenfunction f is nonzero and do es not change signs in Ω( G ) (see [11]). Without loss of generality , w e can assume that f is p ositive on Ω( G ). In this case, we call f a p ositiv e first p -Dirichlet eigenfunction of G . Moreo ver, the first p -Diric hlet eigen v alue is simple in the sense that first p -Diric hlet eigenfunction is unique up to a constan t m ultiple (see [11]). Next, we give the definition of tadp ole graphs. Definition 2.1. The w edge sum of a cycle graph and a path graph on one end v ertex of the path is called a tadpole graph. The other end v ertex of the path is called the end v ertex of the tadp ole graph. The wedge sum v ertex is called the nec k v ertex of the tadp ole graph. The path and the cycle are called the tail and head of the tadpole graph respectively . F or n > i ≥ 3, w e denote the tadp ole graph on n v ertices with the head a cycle of length i as T n,i . More precisely , w e can write T n,i as (2.3) T n,i : t n ∼ t n − 1 ∼ · · · ∼ t i ∼ t i − 1 ∼ · · · ∼ t 2 ∼ t 1 ∼ t i . The path P : t n ∼ t n − 1 ∼ · · · ∼ t i is the tail of T n,i . The cycle C : t i ∼ t i − 1 ∼ · · · ∼ t 2 ∼ t 1 ∼ t i is the head of T n,i . The v ertices t i and t n are the neck v ertex and end vertex of T n,i resp ectiv ely . T adp ole graphs ha v e the follo wing useful sp ectral prop erties. Lemma 2.1. L et n > i ≥ 3 , p > 1 and f b e a p ositive first p -Dirichlet eigen- function of T n,i . Then, f do es not achieve its maximum on the tail of T n,i . Pr o of. W e pro ceed b y con tradiction. Let T n,i b e the tadp ole graph as in (2.3). Supp ose that f ac hiev es its maxim um at some vertex t j with i ≤ j ≤ n . Let e f ∈ R V ( T n,i ) b e suc h that e f ( t k ) =  f ( t j ) 1 ≤ k ≤ j f ( t k ) j < k ≤ n. Then, ∥ f ∥ p p,T n,i = n X k =1 f p ( t k ) ≤ n X k =1 e f p ( t k ) = ∥ e f ∥ p p,T n,i 6 He & Y u and ∥ d f ∥ p p,T n,i = n − 1 X k =1 | f ( t k +1 ) − f ( t k ) | p + | f ( t 1 ) − f ( t i ) | p ≥ n − 1 X k = j | f ( t k +1 ) − f ( t k ) | p = ∥ d e f ∥ p p,T n,i . Therefore, by (2.1), λ 1 ,p ( T n,i ) = R p,T n,i [ f ] ≥ R p,T n,i [ e f ] ≥ λ 1 ,p ( T n,i ) and hence e f is also a first p -Dirichlet eigenfunction of T n,i . Then, 0 <λ 1 ,p ( T n,i ) e f p − 1 ( t 1 ) =∆ p,T n,i e f ( t 1 ) = | e f ( t 1 ) − e f ( t i ) | p − 2 ( e f ( t 1 ) − e f ( t i )) + | e f ( t 1 ) − e f ( t 2 ) | p − 2 ( e f ( t 1 ) − e f ( t 2 )) =0 whic h is ridiculous. This completes the pro of of the lemma. □ W e next show that the first Diric hlet eigenv alue of T n, 4 is greater than that of T n, 3 . Lemma 2.2. F or n ≥ 5 and p > 1 , λ 1 ,p ( T n, 4 ) > λ 1 ,p ( T n, 3 ) . Pr o of. Let f be a p ositiv e first p -Dirichlet eigenfunction of T n, 4 : v n ∼ v n − 1 ∼ · · · ∼ v 4 ∼ v 3 ∼ v 2 ∼ v 1 ∼ v 4 . By symmetry and that λ 1 ,p ( T n, 4 ) is of m ultiplicit y one, w e kno w that (2.4) f ( v 1 ) = f ( v 3 ) . Let m b e a maxim um vertex of f . By Lemma 2.1, m is not in the tail of T n, 4 . So, without loss of generality , m = v 3 or m = v 2 . Let T n, 3 : u n ∼ u n − 1 ∼ · · · ∼ u 3 ∼ u 2 ∼ u 1 ∼ u 3 . When m = v 3 , let e f ∈ R V ( T n, 3 ) b e suc h that e f ( u k ) =  f ( v k ) 3 ≤ k ≤ n f ( m ) 1 ≤ k < 3 . Then, ∥ f ∥ p p,T n, 4 = n X k =1 f p ( v k ) ≤ n X k =1 e f p ( u k ) = ∥ e f ∥ p p,T n, 3 F ab er-Krahn inequality 7 and ∥ d f ∥ p p,T n, 4 = n − 1 X k =1 | f ( v k +1 ) − f ( v k ) | p + | f ( v 1 ) − f ( v 4 ) | p ≥ n − 1 X k =3 | f ( v k +1 ) − f ( v k ) | p = ∥ d e f ∥ p p,T n, 3 . Hence, by (2.1), λ 1 ,p ( T n, 4 ) = R p,T n, 4 [ f ] ≥ R p,T n, 3 [ e f ] > λ 1 ,p ( T n, 3 ) . The last inequality is strict b ecause e f is not a first p -Dirichlet eigenfunction for T n, 3 b y Lemma 2.1. When m = v 2 , let e f ∈ R V ( T n, 3 ) b e suc h that e f ( u k ) = f ( v k ) for k = 1 , 2 , · · · , n . By that 0 <λ 1 ,p ( T n, 4 ) f p − 1 ( v 1 ) =∆ p,T n, 4 f ( v 1 ) = | f ( v 1 ) − f ( v 4 ) | p − 2 ( f ( v 1 ) − f ( v 4 )) + | f ( v 1 ) − f ( v 2 ) | p − 2 ( f ( v 1 ) − f ( v 2 )) ≤| f ( v 1 ) − f ( v 4 ) | p − 2 ( f ( v 1 ) − f ( v 4 )) w e ha v e f ( v 1 ) − f ( v 4 ) > 0. Moreov er, by (2.4), ∥ d f ∥ p p,T n, 4 = n − 1 X k =1 | f ( v k +1 ) − f ( v k ) | p + | f ( v 1 ) − f ( v 4 ) | p > n − 1 X k =1 | f ( v k +1 ) − f ( v k ) | p + | f ( v 3 ) − f ( v 1 ) | p = ∥ d e f ∥ p p,T n, 3 , and it is clear that ∥ f ∥ p p,T n, 4 = ∥ e f ∥ p p,T n, 3 . So, by (2.1), λ 1 ,p ( T n, 4 ) = R p,T n, 4 [ f ] > R p,T n, 3 [ e f ] ≥ λ 1 ,p ( T n, 3 ) . This completes the proof of the lemma. □ Finally , w e compare the first p -Dirichlet eigen v alue of a path graph on n or n + 1 v ertices to that of T n, 3 . W e denote the path graph on n v ertices as P n . 8 He & Y u Lemma 2.3. F or any n ≥ 4 and p > 1 , λ 1 ,p ( P n ) > λ 1 ,p ( P n +1 ) > λ 1 ,p ( T n, 3 ) . Pr o of. Let P n : v n ∼ v n − 1 ∼ · · · ∼ v 2 ∼ v 1 and f be a positive first p -Diric hlet eigenfunction of P n . Let P n +1 : u n ∼ u n − 1 ∼ · · · ∼ u 2 ∼ u 1 ∼ u 0 and e f ( u k ) =  f ( v k ) 1 ≤ k ≤ n 0 k = 0 . Then, by (2.1), λ 1 ,p ( P n ) = R p,P n [ f ] = R p,P n +1 [ e f ] > λ 1 ,p ( P n +1 ) . The last inequalit y is strict b ecause e f is not a p -Diric hlet eigenfunction of P n +1 since e f is not everywhere p ositive in the interior of P n +1 . Moreo v er, let g b e a p ositive first p -Diric hlet eigenfunction of P n +1 and T n, 3 : w n ∼ w n − 1 ∼ · · · ∼ w 3 ∼ w 2 ∼ w 1 ∼ w 3 . When n ≥ 5, let (2.5) e g ( w k ) =  g ( u k ) k ≥ 3 g ( u 3 ) k = 1 , 2 . By that g ( u 3 ) ≥ g ( u 2 ) > g ( u 1 ) > 0, w e hav e ∥ g ∥ p p,P n +1 = n X k =1 g p ( u k ) < n X k =1 e g p ( w k ) = ∥ e g ∥ p p,T n, 3 and ∥ dg ∥ p p,P n +1 = n X k =1 | g ( u k ) − g ( u k − 1 ) | p > n X k =4 | g ( u k ) − g ( u k − 1 ) | p = ∥ d e g ∥ p p,T n, 3 . So, by (2.1), λ 1 ,p ( P n +1 ) = R p,P n +1 ( g ) > R p,T n, 3 ( e g ) ≥ λ 1 ,p ( T n, 3 ) . When n = 4, let e g ( w k ) = g ( u k ) for k = 1 , 2 , 3 , 4 . F ab er-Krahn inequality 9 Note that g ( u 3 ) = g ( u 1 ) by symmetry . So, ∥ dg ∥ p p,P 5 = 4 X k =1 | g ( u k ) − g ( u k − 1 ) | p > 4 X k =2 | g ( u k ) − g ( u k − 1 ) | p = 4 X k =2 | e g ( w k ) − e g ( w k − 1 ) | p + | e g ( w 1 ) − e g ( w 3 ) | p = ∥ d e g ∥ p p,T 4 , 3 , and it is clear that ∥ g ∥ p p,P 5 = ∥ e g ∥ p p,T 4 , 3 . Th us, b y (2.1), λ 1 ,p ( P 5 ) = R p,P 5 [ g ] > R p,T 4 , 3 [ e g ]) ≥ λ 1 ,p ( T 4 , 3 ) . This completes the proof of the lemma. □ W e w oul like to men tion that the sp ectral prop erties ab ov e for tadp ole graphs are mainly given in Katsuda-Urak a w a [13] for normalized combinatorial Laplacian and the arguments ab o v e are similar to that of Katusda-Urak aw a [13]. Here, w e just mo dify them for the un-normalized com binatorial p -Laplacian. 3. Pr oofs of main resul ts In this section, w e prov e the main results. W e first prov e Theorem 1.2. Pr o of of The or em 1.2. Let f b e a p ositive first p -Diric hlet eigenfunction of G . Supp ose that f ( m ) = max v ∈ V ( G ) f ( v ) . Let P : v n ∼ v n − 1 ∼ · · · ∼ v i = m b e a shortest path joining the b oundary vertex v n to m = v i and supp ose that v 1 , v 2 , · · · , v n are the v ertices of G . Because f ( m ) > 0 and P is a shortest path joining v n and v i , we know that i ≥ 2. When i ≥ 3, let T n, 3 : u n ∼ u n − 1 ∼ · · · ∼ u i ∼ · · · ∼ u 3 ∼ u 2 ∼ u 1 ∼ u 3 , and e f ( u k ) =  f ( v k ) i ≤ k ≤ n f ( m ) 1 ≤ k < i. 10 He & Y u Then, ∥ f ∥ p p,G = n X k =1 f p ( v k ) ≤ n X k =1 e f p ( u k ) = ∥ e f ∥ p p,T n, 3 and ∥ d f ∥ p p,G ≥ n − 1 X k = i | f ( v k +1 ) − f ( v k ) | p = n − 1 X k = i | e f ( v k +1 ) − e f ( v k ) ∥ p = ∥ d e f ∥ p p,T n, 3 . So, by (2.1), λ 1 ,p ( G ) = R p,G [ f ] ≥ R p,T n, 3 [ e f ] > λ 1 ,p ( T n, 3 ) . The last inequalit y is strict b ecause e f is not a p ositive first p -Dirichlet eigenfunc- tion of T n, 3 b y Lemma 2.1. When i = 2, w e know that v 1 ∼ v 2 in G b ecause v 2 is not a b oundary vertex. Moreo v er, v 1 can not b e adjacen t to v j with j ≥ 5 b ecause P is the shortest path joining v n to v 2 . Then, by the adjacency of v 1 to v 3 and v 4 , we only ha v e the follo wing four cases: (1) v 1 ∼ v 3 and v 1 ∼ v 4 : In this case, G = P n , by Lemma 2.3, λ 1 ,p ( G ) > λ 1 ,p ( T n, 3 ) . (2) v 1 ∼ v 3 and v 1 ∼ v 4 : In this case, G = T n, 4 , by Lemma 2.2, λ 1 ,p ( G ) > λ 1 ,p ( T n, 3 ) . (3) v 1 ∼ v 3 and v 1 ∼ v 4 : In this case, b y that 0 <λ 1 ( G ) f p − 1 ( v 1 ) =∆ p,G f ( v 1 ) = | f ( v 1 ) − f ( v 2 ) | p − 2 ( f ( v 1 ) − f ( v 2 )) + | f ( v 1 ) − f ( v 3 ) | p − 2 ( f ( v 1 ) − f ( v 3 )) + | f ( v 1 ) − f ( v 4 ) | p − 2 ( f ( v 1 ) − f ( v 4 )) ≤| f ( v 1 ) − f ( v 3 ) | p − 2 ( f ( v 1 ) − f ( v 3 )) + | f ( v 1 ) − f ( v 4 ) | p − 2 ( f ( v 1 ) − f ( v 4 )) , w e ha v e f ( v 1 ) − f ( v 3 ) > 0 or f ( v 1 ) − f ( v 4 ) > 0. If f ( v 1 ) − f ( v 3 ) > 0, let G ′ = G − { v 1 , v 3 } . It is clear that G ′ = T n, 4 . So, by (2.1) and Lemma 2.2, λ 1 ,p ( G ) = R p,G [ f ] > R p,G ′ [ f ] ≥ λ 1 ,p ( G ′ ) = λ 1 ,p ( T n, 4 ) > λ 1 ,p ( T n, 3 ) . If f ( v 1 ) − f ( v 4 ) > 0, let G ′ = G − { v 1 , v 4 } . Then G ′ = T n, 3 . So, b y (2.1), λ 1 ,p ( G ) = R p,G [ f ] > R p,G ′ [ f ] ≥ λ 1 ,p ( G ′ ) = λ 1 ,p ( T n, 3 ) . (4) v 1 ∼ v 3 and v 1 ∼ v 4 : In this case, G = T n, 3 . This completes the proof of the theorem. □ Finally , w e pro v e Theorem 1.3. F ab er-Krahn inequality 11 Pr o of of The or em 1.3. If G is a path graph, by Lemma 2.3, λ 1 ,p ( G ) = λ 1 ,p ( P n +1 ) > λ 1 ,p ( T n, 3 ) . So, we assume that G is not a path graph and hence there exists a v ertex v ∈ V ( G ) suc h that deg v ≥ 3. Then, b y that 2 n > X x ∈ Ω( G ) deg( x ) = deg ( v ) + X Ω( G ) \{ v } deg( x ) ≥ 3 + 2( | Ω( G ) | − 1) , w e ha v e (3.1) | Ω( G ) | ≤ n − 1 . Let f b e a p ositiv e first p -Diric hlet eigenfunction of G and m ∈ V ( G ) b e a maxim um point of f . Let P : v n ∼ v n − 1 ∼ · · · ∼ v i = m b e a shortest path joining the b oundary vertex v n to v i = m . Note that if | E ( G ) | − | E ( P ) | = 1, then G = P n +1 b ecause m is not b oundary vertex and P is a shortest path joining v n to m . Th us | E ( G ) | − | E ( P ) | ≥ 2. When | E ( G ) | − | E ( P ) | ≥ 3, w e hav e i ≥ 3. Let T n, 3 : u n ∼ u n − 1 ∼ · · · ∼ u i ∼ · · · ∼ u 3 ∼ u 2 ∼ u 1 ∼ u 3 and e f ( u k ) =  f ( v k ) i ≤ k ≤ n f ( m ) 1 ≤ k < i. Then, by (3.1), we ha ve ∥ f ∥ p p,G = n − 1 X k = i f p ( v k ) + X x ∈ Ω( G ) \{ v i ,v i +1 , ··· ,v n − 1 } f p ( x ) ≤ n − 1 X k = i e f p ( u k ) + ( | Ω( G ) | − ( n − i )) f p ( m ) ≤ n − 1 X k = i e f p ( u k ) + ( i − 1) f p ( m ) = ∥ e f ∥ p p,T n, 3 and ∥ d f ∥ p p,G ≥ n − 1 X k = i | f ( v k +1 ) − f ( v k ) | p = n − 1 X k = i | e f ( u k +1 ) − e f ( u k ) | p = ∥ d e f ∥ p p,T n, 3 . So, by (2.1), (3.2) λ 1 ,p ( G ) = R p,G [ f ] ≥ R p,T n, 3 [ e f ] > λ 1 ,p ( T n, 3 ) . 12 He & Y u The last inequalit y is strict b ecause e f is not a p ositive first p -Dirichlet eigenfunc- tion of T n, 3 b y Lemma 2.1. When | E ( G ) | − | E ( P ) | = 2, we ha v e i = 2. Because v 2 is not a b oundary vertex and P is a shortest path joining v n and v 2 , there is another v ertex v 1 adjacen t to v 2 . If there is no other vertex of G , then G is a connected graph on n vertices with nonempty b oundary . By Theorem 1.2, we kno w that λ 1 ,p ( G ) ≥ λ 1 ,p ( T n, 3 ) with equalit y if and only if G = T n, 3 . Otherwise, let v 0 b e another vertex of G whic h should b e one of the end p oints of the remaining edge. Because G  = P n +1 , v 0 ∼ v j for some j = 2 , 3 , · · · , n − 1. Let G ′ = G − { v 0 } . Then G ′ = P n and b y (2.1) and Lemma 2.3, λ 1 ,p ( G ) = R p,G [ f ] > R p,G ′ [ f ] ≥ λ 1 ,p ( P n ) > λ 1 ,p ( T n, 3 ) . This completes the proof of the theorem. □ Finally , w e come to pro v e Theorem 1.4. Pr o of of The or em 1.4. F or an y nonempt y subset U of Ω( G ), note that (3.3) 2 n > X x ∈ U deg( x ) ≥ 2 | U | . So, | U | ≤ n − 1. Thus, (3.4) | E ( U, U c ) | | U | ≥ 1 | U | ≥ 1 n − 1 , and λ 1 , 1 ( G ) = h D ( G ) ≥ 1 n − 1 . If the equalit y holds, let U ⊂ Ω( G ) b e such that | E ( U, U c ) | | U | = h D ( G ) = 1 n − 1 . Then, b y (3.3) and (3.4), we kno w that | E ( U, U c ) | = 1 and | U | = n − 1. Then, b y that 1 ≤| B ( G ) | =2 n − X x ∈ Ω( G ) deg( x ) ≤ 2 n − 2(Ω( G ) − | U | ) − X x ∈ U deg( x ) ≤ 2 − 2(Ω( G ) − | U | ) . (3.5) W e ha v e U = Ω( G ) and G has at most t w o p endan t v ertices. F ab er-Krahn inequality 13 If | B ( G ) | = 2, then by (3.5), deg( x ) = 2 for any x ∈ Ω( G ). This implies that G = P n +1 . How ev er, h D ( P n +1 ) = 2 n − 2 > 1 n − 1 . Th us, | B ( G ) | = 1. By (3.5), 2 n − 2 ≤ X x ∈ Ω( G ) deg( x ) = 2 n − 1 . 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Dep ar tment of Ma thema tics, Shantou University, Shantou, Guangdong, 515063, China Email addr ess : 18wkhe@stu.edu.cn Dep ar tment of Ma thema tics, Shantou University, Shantou, Guangdong, 515063, China Email addr ess : cjyu@stu.edu.cn