An extension of Katsuda-Urakawa's Faber-Krahn inequality

AN EXTENSION OF KA TSUD A-URAKA W A’S F ABER-KRAHN INEQUALITY W ANKAI HE AND CHENGJIE YU 1 Abstract. In this pap er, motiv ated by our previous work [7], we prov e that the minim um of the first Dirichlet eigen v alues for the normalized com binatorial p -Laplacian on connected finite graphs with boundary consisting of n edges is only ac hiev ed b y the tadp ole graph T n, 3 . This result extends the F aber-Krahn inequalit y of Katsuda-Urak a wa [9] to normalized com binatorial p -Laplacians. Our argumen t is m uch simpler than that of Katsuda-Urak aw a. 1. Intr oduction In [9], Katsuda and Urak aw a obtained the following sharp F ab er-Krahn inequal- it y for the fist Dirichlet eigen v alues of the normalized combinatorial Laplacian on graphs with a fixed n umber of edges. Theorem 1.1 (Katsuda-Urak a w a [9]) . L et G b e a c onne cte d gr aph with b oundary that c onsists of n ≥ 4 e dges. Then, λ 1 ( G ) ≥ λ 1 ( T n, 3 ) and the e quality holds if and only if G = T n, 3 . Here the p endan t v ertices of G are considered as the b oundary v ertices of G , T n, 3 means the tadp ole graph on n -vertices with the head a cycle of length 3 (see the next section for details), and λ 1 ( G ) is the first Diric hlet eigen v alue of the normalized combinatorial Laplacian on G : ∆ G f ( x ) = 1 deg( x ) X y ∼ x ( f ( x ) − f ( y )) . The pro of of Theorem 1.1 in [9] is rather complicated using three kinds of surgeries to reduce the first Diric hlet eigen v alue of a graph with b oundary . In this 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, normalized combina- torial Laplacian . 1 Researc h partially supported by GDNSF with contract no. 2025A1515011144 and 2026A1515012267. 1 2 He & Y u pap er, motiv ated by our previous work [7], we extend Theorem 1.1 to normalized com binatorial p -Laplacian: ∆ p,G f ( x ) = 1 deg( x ) X y ∼ x | f ( x ) − f ( y ) | p − 2 ( f ( x ) − f ( y )) . Our argumen t is muc h simpler than that of Katsuda-Urak aw a [9] b y just using a simple surgery on the graph. Theorem 1.2. 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 ) and the e quality holds if and only if G = T n, 3 . When p = 1, the F ab er-Krahn inequality in Theorem 1.2 still holds by that λ 1 , 1 ( G ) = lim p → 1 +1 λ 1 ,p ( G ) . (see [5] for example.) Ho w ever, the rigidit y part of the F ab er-Krahn inequality do es not hold. In fact, b y noting that λ 1 , 1 ( G ) = h D ( G ) where h D ( G ) is the Dirichlet Cheeger constant (see (2.3) for definition), one is not hard to see that λ 1 , 1 ( G ) ≥ 1 2 n − 1 with equality if and only if G has only one p endan t v ertex. The classical F aber-Krahn inequality (see [2, P . 87]) sa ys that λ 1 (Ω) ≥ λ 1 ( B ) where Ω is an y smooth bounded Euclidean domains and B is a ball of the same v olume as Ω, and the equalit y holds if and only if Ω is the translation of the ball B . So, Theorem 1.1 is a discrete version of F aber-Krahn inequality . Discrete analogues of the classical F ab er-Krahn inequalit y w ere first considered in the pioneer work [3] of F riedman. In [3], F riedman formulated the problem of finding the F aber-Krahn inequalit y for domains in a homogeneous tree with fixed total length and conjectured that the minimum is achiev ed by a geo desic ball in the homogeneous tree. Later, Pruss [13] dispro v ed F riedman’s conjecture, and Leydold [10, 11] completely solv ed F riedman’s problem. In [4], F riedman estab- lished F aber-Krahn inequalities for eigen v alues of the com binatorial Laplacian on graphs with a fixed n um b er of v ertices, and the second named author and Ying- tao Y u [15] extended the result to Steklov eigen v alues. The general problem of finding graphs satisfying the so called F aber-Krahn prop ert y in certain class of graphs was introduced b y Bıyıko˘ glu and Leydold [1]. In [1, 16, 18, 17, 14, 12], the authors solved the extremum problem form ulated in [1] for v arious classes of graphs. Katsuda-Urak aw a’s F aber-Krahn inequality 3 2. Pr oof of the main resul t W e first in tro duce some notations and basic facts. F or a nontrivial connected finite graph G , we denote the collection of p endan t v ertices of G as B ( G ) which is viewed as the b oundary of G . The set Ω( G ) := V ( G ) \ B ( G ) is view ed as the interior of G . F or p > 1, the normalized combinatorial p - Laplacian op erator on G is defined as ∆ p,G f ( x ) = 1 deg( x ) X y ∼ x | f ( x ) − f ( y ) | p − 2 ( f ( x ) − f ( y )) , ∀ x ∈ V ( G ) , where f ∈ R V ( G ) . A real n um b er λ is called a p -Dirichlet eigenv alue of G if the follo wing 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 , and f is called a p -Diric hlet eigenfunction of G . The smallest p -Dirichlet eigenv alue of G is denoted as λ 1 ,p ( G ) which is called the first p -Diric hlet eigen v alue of G , and the corresp onding eigenfunction is called a first p - Diric hlet eigenfunction of G . The first p -Diric hlet eigenv alue can b e characterized b y the minimum of p -Ra yleigh quotien t: (2.1) λ 1 ( G ) = min f ∈ C B ( G ) \{ 0 } R p,G [ f ] , and the minimum is only achiev ed by first 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 ) deg ( x ) = X x ∈ Ω( G ) | f | p ( x ) deg ( x ) since f ∈ C B ( G ) . If B ( G )  = ∅ and p > 1, as shown in [8], the first p -Dirichlet eigenv alue λ 1 ,p ( G ) is positive and the first p -Diric hlet eigenfunction f is nonzero and do es not change signs in Ω( G ). Without loss of generalit y , we can assume that f is p ositiv e on Ω( G ). In this case, we call f a positive first p -Diric hlet eigenfunction of G . 4 He & Y u Moreo v er, as sho wn in [8], λ 1 ( G ) is of multiplicit y one in the sense that the first p -Diric hlet eigenfunction is unique up to a constant multiple. When p = 1, the definition of the normalized combinatorial 1-Laplacian for a graph is subtle, see [2] for example. How ever, the first 1-Diric hlet eigen v alue λ 1 , 1 ( G ) can b e also c haracterized b y Rayleigh quotient in (2.1) with p = 1. In fact, it is well-kno wn (see [6, 8] for example) that (2.2) λ 1 , 1 ( G ) = h D ( G ) where (2.3) h D ( G ) = inf ∅ = U ⊂ Ω( G ) | E ( U, U c ) | P x ∈ U deg( x ) . By setting f = 1 U with U a nonempty subset in Ω( G ) in (2.1), one has λ 1 ,p ( G ) ≤ h D ( G ) . Moreo v er, b y setting U = { x } where x is an interior v ertex of G , one has h D ( G ) ≤ 1 and (2.4) λ 1 ,p ( G ) ≤ 1 . Next, w e in tro duce the notion of tadp ole graph. F or n > i ≥ 3, we denote the tadp ole graph on n vertices with the head a cycle of length i as T n,i whic h can represen ted as (2.5) 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 called the tail of T n,i . The cycle C : t i ∼ t i − 1 ∼ · · · ∼ t 2 ∼ t 1 ∼ t i is called the head of T n,i . The vertices t i and t n are called the nec k vertex 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 with their pro ofs are the same as those in [7]. See also [9]. 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 . Lemma 2.2. F or n ≥ 5 and p > 1 , λ 1 ,p ( T n, 4 ) > λ 1 ,p ( T n, 3 ) . Katsuda-Urak aw a’s F aber-Krahn inequality 5 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 ) . Her e P n is the p ath gr aph on n vertic es. W e are no w ready to prov e Theorem 1.2. Pr o of. Let f b e a positive first p -Dirichlet function of G and m ∈ V ( G ) be a maxim um p oin t of f . Let P : v n ∼ v n − 1 ∼ · · · ∼ v i = m b e a shortest path joining a b oundary v ertex v n to m . Then, 2 n = X x ∈ B ( G ) deg( x ) + n − 1 X k = i +1 deg( v k ) + X x ∈ Ω( G ) \{ v i +1 , ··· ,v n − 1 } deg( x ) = | B ( G ) | + 2( n − i − 1) + n − 1 X k = i +1 (deg( v k ) − 2) + X x ∈ Ω( G ) \{ v i +1 , ··· ,v n − 1 } deg( x ) . So, n − 1 X k = i +1 (deg( v k ) − 2) + X x ∈ Ω( G ) \{ v i +1 , ··· ,v n − 1 } deg( x ) = 2( i + 1) − | B ( G ) | ≤ 2 i + 1 . (2.6) When | E ( G ) | − | E ( P ) | ≥ 3, we ha v 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 ( v i ) 1 ≤ k < i. Then, by (2.6), ∥ d f ∥ p p,G = n − 1 X k = i +1 f p ( v k ) deg ( v k ) + X x ∈ Ω( G ) \{ v i +1 , ··· ,v n − 1 } f p ( x ) deg ( x ) ≤ 2 n − 1 X k = i +1 f p ( v k ) + f p ( m )   n − 1 X k = i +1 (deg( v k ) − 2) + X x ∈ Ω( G ) \{ v i +1 , ··· ,v n − 1 } deg( x )   ≤ 2 n − 1 X k = i +1 f p ( v k ) + (2 i + 1) f p ( m ) = ∥ e f ∥ p p,T n, 3 6 He & Y u 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 . Hence, by (2.1), (2.7) 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 ositiv e first p -Diric hlet eigenfunc- tion of T n, 3 b y Lemma 2.1. When | E ( G ) | − | E ( P ) | = 2, we hav 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 v ertex of G , then the remaining edge of G should b e { v 1 ∼ v j } for some j = 3 , 4 , · · · , n − 1. Because P is a shortest path joining v n and v 2 , we know that j = 3 or j = 4. When j = 3, G = T n, 3 and w e are done. When j = 4, G = T n, 4 . By Lemma 2.2, λ 1 ,p ( G ) = λ 1 ,p ( T n, 4 ) > λ 1 ,p ( T n, 3 ) . Otherwise, let v 0 b e another v ertex of G whic h should b e one of the end p oin ts of the remaining edge. Supp ose v 0 is adjacent to v i for some i = 1 , 2 , · · · , n − 1. When v 0 ∼ v 1 , G = P n +1 . By Lemma 2.3, λ 1 ,p ( G ) = λ 1 ,p ( P n +1 ) > λ 1 ( T n, 3 ) . When v 0 ∼ v j for some j = 2 , 3 , · · · , n − 1. Let G ′ = G − { v 0 } . Then, G ′ = P n . Moreo v er ∥ d f ∥ p p,G ′ = ∥ d f ∥ p p,G − f p ( v j ) and ∥ f ∥ p p,G ′ = ∥ f ∥ p p,G − f p ( v j ) . By (2.4), ∥ d f ∥ p p,G ∥ f ∥ p p,G = λ 1 ,p ( G ) ≤ 1 , w e ha v e R p,G ′ [ f ] = ∥ d f ∥ p p,G − f p ( v j ) ∥ f ∥ p p,G − f p ( v j ) ≤ λ 1 ,p ( G ) . So, by Lemma 2.1 and Lemma 2.3, λ 1 ,p ( G ′ ) = R p,G [ f ] ≥ R p,G ′ [ e f ] ≥ λ 1 ,p ( P n ) > λ 1 ,p ( T n, 3 ) . Finally , when | E ( G ) | − | E ( P ) | = 1, it is clear that G = P n +1 . Then, by Lemma 2.3, λ 1 ,p ( P n +1 ) > λ 1 ,p ( T n, 3 ) . This completes the pro of of the Theorem. □ Katsuda-Urak aw a’s F aber-Krahn inequality 7 References [1] Bıyıko˘ glu T., Leydold J., F aber-Krahn type inequalities for trees. J. Combin. Theory Ser. B 97 (2007), no. 2, 159–174. [2] Chav el I., Eigenv alues in Riemannian geometry . Including a chapter b y Burton Randol. With an app endix b y Jozef Do dziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. [3] F riedman J., Some geometric asp ects of graphs and their eigenfunctions. Duke Math. J. 69 (1993), no. 3, 487–525. [4] F riedman J., Minimum higher eigenv alues of Laplacians on graphs. Duke Math. J. 83 (1996), no. 1, 1–18. 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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