On the spectral radius, energy and Estrada index of the Sombor matrix of graphs
aa r X i v : . [ m a t h . C O ] M a r On the spectral radius, energy and Estrada index ofthe Sombor matrix of graphs ∗ Zhen Lin † , Lianying Miao School of Mathematics, China University of Mining and Technology,Xuzhou, 221116, Jiangsu, P.R. China
Abstract
Let G be a simple undirected graph with vertex set V ( G ) = { v , v , . . . , v n } andedge set E ( G ). The Sombor matrix S ( G ) of a graph G is defined so that its ( i, j )-entry is equal to q d i + d j if the vertices v i and v j are adjacent, and zero otherwise,where d i denotes the degree of vertex v i in G . In this paper, lower and upper boundson the spectral radius, energy and Estrada index of the Sombor matrix of graphsare obtained, and the respective extremal graphs are characterized. Mathematics Subject Classification 2010:
Keywords:
Sombor matrix; Sombor spectral radius; Sombor energy;Sombor Estrada index Introduction
Let G be a simple undirected graph with vertex set V ( G ) and edge set E ( G ). For v ∈ V ( G ), N G ( v ) denotes the neighborhood of v in G , and d v = | N G ( v ) | denotes thedegree of vertex v in G . The minimum and maximum degree of a vertex in G are denotedby δ and ∆, respectively.Given a graph G , the adjacency matrix A is the n × n matrix whose ( i, j )-entry is 1if v i v j ∈ E ( G ) and zero otherwise. Thus A is a real symmetric matrix, its eigenvaluesmust be real and arranged in non-increasing order λ ≥ λ ≥ · · · ≥ λ n , where λ is called ∗ Supported by the National Natural Science Foundation of China (No. 12071411, 11771443). † Corresponding author. E-mail addresses: [email protected] (Z. Lin), [email protected](L. Miao). G . For the adjacency spectra, one may refer to [24, 27, 28] and thereferences therein.The energy of G , introduced by Gutman [6], is defined as E A ( G ) = n P i =1 | λ i | , which isintensively studied in chemistry, since it can be used to approximate the total π -electronenergy of a molecule, see for example [7, 8]. There is a wealth of literature relating to theenergy see, for example, [9] for surveys, and see [22], for monograph.In 2021, a new vertex-degree-based molecular structure descriptor was put forward byGutman [10], the Sombor index of a graph G , defined as SO ( G ) = P uv ∈ E ( G ) p d u + d v .The study of the Sombor index of graphs has quickly received much attention. Cruz et al.[2] studied the Sombor index of chemical graphs, and characterized the graphs extremalwith respect to the Sombor index over the following sets: (connected) chemical graphs,chemical trees, and hexagonal systems. Deng et al. [4] obtained a sharp upper bound forthe Sombor index among all molecular trees with fixed numbers of vertices, and charac-terized those molecular trees achieving the extremal value. Das et al. [3] gave lower andupper bounds on the Sombor index of graphs by using some graph parameters. Moreover,they obtained several relations on Sombor index with the first and second Zagreb indicesof graphs. R´eti et al. [26] characterized graphs with the maximum Sombor index in theclasses of all connected unicyclic, bicyclic, tricyclic, tetracyclic, and pentacyclic graphsof a fixed order. Redˇzepovi´c [25] showed that the Sombor index has good predictivepotential. For other related results, one may refer to [11, 18, 20, 30] and the referencestherein.The aim of this paper is to study the Sombor index from an algebraic viewpoint, whichis a natural idea in mathematical chemistry. The Sombor matrix S ( G ) of a graph G isdefined so that its ( i, j )-entry is equal to q d i + d j if the vertices v i and v j are adjacent,and zero otherwise, where d i denotes the degree of vertex v i in G . The eigenvalues of S ( G )are denoted by ρ ( G ) ≥ ρ ( G ) ≥ · · · ≥ ρ n ( G ), where ρ ( G ) is called the Sombor spectralradius of G . The Sombor energy and Estrada index of the Sombor matrix of graphs aredefined as E ( G ) = P ni =1 | ρ i ( G ) | and EE ( G ) = P ni =1 e ρ i ( G ) , respectively. We obtain lowerand upper bounds on the Sombor spectral radius, Sombor energy and Sombor Estradaindex of graphs, and characterize the respective extremal graphs.2 Preliminaries
The diameter of a graph G , denoted by diam ( G ), is the maximum distance between anypair of vertices of G . Let K s, t and K n denote the complete bipartite graph with s + t vertices and the complete graph with n vertices, respectively. The first Zagreb index [14] Z and forgotten topological index [5] F of G are defined as Z = Z ( G ) = n X i =1 d i = X v i v j ∈ E ( G ) ( d i + d j ) , F = F ( G ) = n X i =1 d i = X v i v j ∈ E ( G ) ( d i + d j ) . Lemma 2.1 ([1, 32])
Let G be a graph with n vertices. Then r Z n ≤ λ ≤ ∆ . The equality in the left hand side holds if and only if G is regular or semiregular. If G is aconnected graph, then the equality in the right hand side holds if and only if G is regular. Lemma 2.2 ([1, 16])
Let G be a connected graph of order n with m edges. Then mn ≤ λ ≤ √ m − n + 1 . The equality in the left hand side holds if and only if G is a regular graph, and the equalityin the right hand side holds if and only if G ∼ = K , n − or G ∼ = K n . Lemma 2.3 ([15])
Let A be a symmetric matrix of order n with eigenvalues ξ ≥ ξ ≥· · · ≥ ξ n , and let B be its principal submatrix with eigenvalues η ≥ η ≥ · · · ≥ η k and n > k . Then ξ i ≥ η i ≥ ξ n − k + i for i = 1 , , . . . , k . Lemma 2.4 ([19])
Let a ≥ a ≥ · · · ≥ a n ≥ be a sequence of non-negative realnumbers. Then n X i =1 a i + n ( n − n Y i =1 a i ! /n ≤ n X i =1 √ a i ! ≤ ( n − n X i =1 a i + n n Y i =1 a i ! /n . Lemma 2.5 ([12])
Let M be an n × n non-negative symmetric matrix such that itsunderlying graph is connected. Let λ ( M ) , λ ( M ) , . . . , λ k ( M ) be all the eigenvalues of M with absolute value equal to λ ( M ) . Then k > if and only if all closed walks in G havelength divisible by k . emma 2.6 ([13, 22]) Let T n be a tree with n ≥ vertices. Then √ n − E A ( K , n − ) ≤ E A ( T n ) ≤ E A ( P n ) = π n + 1) − , if n is even , π n + 1) − , if n is odd . Lemma 2.7 If G is a connected graph with k ≥ distinct Sobmor eigenvalues, then diam ( G ) ≤ k − . Proof.
Let S be the Sombor matrix of G and ρ > ρ > · · · > ρ k be its k distinctSombor eigenvalues. Let X be the unit (column) eigenvector corresponding the largesteigenvalues ρ . Then X is a positive vector. From Theorem 2.1 in [21], it follows that k Y i =2 ( S − ρ i I ) = S k − + c S k − + c k − S + c k − I = k Y i =2 ( ρ − ρ i ) XX T = M. Observe that ( M ) ij > i, j = 1 , , . . . , n . Therefore, for i = j , there is a positiveinteger l with 1 ≤ l ≤ k − S l ) ij >
0, which implies that there is a path oflength l between v i and v j , that is, diam ( G ) ≤ k −
1. The proof is completed. ✷ Lemma 2.8
Let G be a graph with n vertices. Then | ρ | = | ρ | = · · · = | ρ n | if and onlyif G ∼ = K n or G ∼ = n K . Proof.
First we assume that | ρ | = | ρ | = · · · = | ρ n | . Let t be the number of isolatedvertices in G . If t ≥
1, then ρ = ρ = · · · = ρ n = 0 and hence G ∼ = K n . Otherwise, t = 0. If ∆ = 1, then d = d = · · · = d n = 1 and hence G ∼ = n K . Otherwise,∆ ≥
2. Then G contains a connected component H with at least 3 vertices. If H is acomplete graph with p ≥ | ρ ( H ) | = √ p − > √ p −
1) = | ρ ( H ) | , acontradiction. Otherwise, H is not a complete graph. By Lemma 2.3, ρ ( H ) ≥
0. By thePerron-Frobenius theorem, ρ ( H ) > ρ ( H ), a contradiction. Conversely, one can easilycheck that | ρ | = | ρ | = · · · = | ρ n | holds for K n and n K . This completes the proof. ✷ Lemma 2.9
Let G be a graph with Sombor eigenvalues ρ , ρ , . . . , ρ n . Then (i) n P i =1 ρ i = 0 , n P i =1 ρ i = − P ≤ i Theorem 3.1 Let G be a graph with n vertices, the maximum degree ∆ and minimumdegree δ . Then √ δλ ≤ ρ ≤ √ λ (3 . ith equality if and only if G is a regular graph. Proof. Firstly, we prove the left hand side of (3.1). Let X = ( x , x , . . . , x n ) be a uniteigenvector of G corresponding to λ . By the Rayleigh-Ritz Theorem, we have ρ ≥ X T S X ≥ X v i v j ∈ E ( G ) q d i + d j x i x j ≥ √ δ X v i v j ∈ E ( G ) x i x j = √ δλ . (3 . d i = d j for each edge v i v j ∈ E ( G ), that is, G is regular.Clearly, if G is a regular graph, the equality in the left hand side of (3.1) holds.Secondly, we prove the right hand side of (3.1). Let Y = ( y , y , . . . , y n ) be a uniteigenvector of G corresponding to ρ . By the Rayleigh-Ritz Theorem, we have λ ≥ Y T AY ≥ X v i v j ∈ E ( G ) y i y j . Similarly, we have ρ = Y T S Y = 2 X v i v j ∈ E ( G ) q d i + d j y i y j ≤ √ X v i v j ∈ E ( G ) y i y j ≤ √ λ . (3 . d i = d j for any v i v j ∈ E ( G ), that is G is a regulargraph. Conversely, it is easy to verify that the equality in the right hand side of (3.1)holds when G is a regular graph.Combining the above arguments, we have the proof. ✷ By Lemmas 2.1, 2.2 and Theorem 3.1, we have the following corollaries. Corollary 3.2 Let G be a connected graph with n vertices, m edges, the maximum degree ∆ and minimum degree δ . Then δ r Z n ≤ ρ ≤ √ . The equality holds if and only if G is a regular graph. orollary 3.3 Let G be a connected graph with n vertices, m edges, the maximum degree ∆ and minimum degree δ . Then √ mδn ≤ ρ ≤ ∆ √ m − n + 2 . The equality in the left hand side holds if and only if G is a regular graph, and the equalityin the right hand side holds if and only if G ∼ = K n . Theorem 3.4 Let G be a graph with n vertices. Then r Fn ≤ ρ ≤ r n − Fn . (3 . The equality in the left hand side holds if and only if G ∼ = K n or G ∼ = n K . If G is aconnected graph, then the equality in the right hand side holds if and only if G ∼ = K n . Proof. By Lemma 2.9, we have nρ ≥ F . The equality holds if and only if | ρ | = | ρ | = · · · = | ρ n | , by Lemma 2.8, we have the result. By the Cauchy-Schwarz inequality, we have ρ = 2 F − n X i =2 ρ i ≤ F − n − n X i =2 ρ i ! = 2 F − n − ρ . Thus ρ ≤ q n − Fn . The equality holds if and only if ρ = ρ = · · · = ρ n , by Lemma2.7, we have diam ( G ) = 1, that is G ∼ = K n . Conversely, if G ∼ = K n , then S ( G ) = √ n − A ( G ). Thus ρ ( G ) = √ n − , ρ ( G ) = · · · = ρ n ( G ) = −√ n − ✷ Theorem 3.5 Let uv be an edge of a connected graph G with d u ≥ and d v ≥ .Let X be a unit eigenvector of G corresponding to ρ ( G ) , and let N G ( u ) ∩ N G ( v ) = Φ and G ∗ = G − { vw : w ∈ N G ( v ) \{ u }} + { uw : w ∈ N G ( v ) \{ u }} . If x u ≥ x v , then ρ ( G ∗ ) > ρ ( G ) . Proof. Since X is a unit eigenvector of ρ ( G ), X is positive if G is connected. From thehypothesis, we have ρ ( G ∗ ) − ρ ( G ) ≥ X T S ( G ∗ ) X − X T S ( G ) X X p ∈ N G ( u ) \{ v } ( q d p + ( d u + d v − − q d p + d u ) x u x p +2 X w ∈ N G ( v ) \{ u } ( p d w + ( d u + d v − x u − p d w + d v x v ) x w +2( p + ( d u + d v − − p d u + d v ) x u x v ≥ X p ∈ N G ( u ) \{ v } ( q d p + ( d u + d v − − q d p + d u ) x u x p +2 X w ∈ N G ( v ) \{ u } ( p d w + ( d u + d v − − p d w + d v ) x v x w +2( p + ( d u + d v − − p d u + d v ) x u x v > . The proof is completed. ✷ Corollary 3.6 Let T n be a tree with n vertices. Then ρ ( T n ) ≤ p ( n − n − n + 2) with equality if and only if G = K , n − . Li and Wang [17] showed that the path P n is uniquely the tree on n ≥ Theorem 3.7 ([17]) Let T n be a tree with n ≥ vertices. Then ρ ( P n ) ≤ ρ ( T n ) ≤ ρ ( K , n − ) . The equality in the left hand side holds if and only if T n ∼ = P n , and the equality in theright hand side holds if and only if T n ∼ = K , n − . Problem 3.8 For a given class of graphs, characterize the graphs with the maximum orminimum Sombor spectral radius. On the Sombor energy Theorem 4.1 Let G be a graph with n vertices. Then q F + n ( n − 1) ( | det S ( G ) | ) /n ≤ E ( G ) ≤ q n − F + n ( | det S ( G ) | ) /n . roof. Replacing a i = ρ i ( G ) in Lemma 2.4, we have the proof. ✷ Theorem 4.2 Let G be a graph with n ≥ vertices. Then √ F ≤ E ( G ) ≤ √ nF . (4 . The equality in the right hand side holds if and only if G ∼ = K n or G ∼ = n K . If G isa connected graph, then the equality in the left hand side holds if and only if G ∼ = K s, t , s + t = n . Proof. By the Cauchy-Schwarz inequality, we have E ( G ) = n X i =1 | ρ i | ≤ vuut n n X i =1 ρ i = √ nF . with equality if and only if | ρ | = | ρ | = · · · = | ρ n | . By Lemma 2.8, we have G ∼ = K n or G ∼ = n K . Since n P i =1 ρ i = − P ≤ i Let G be a complete bipartite graph with n vertices. Then p ( n − n − n + 2) ≤ E ( G ) ≤ rl n m j n k + j n k l n m . The equality in the left hand side holds if and only if G ∼ = K , n − , and the equality in theright hand side holds if and only if G ∼ = K ⌈ n ⌉ , ⌊ n ⌋ . roof. By the proof of Theorem 4.2, we have E ( K s, t ) = 2 √ s t + st = 2 p ( n − t ) t + ( n − t ) t , ≤ t ≤ j n k . Let f ( x ) = 2 p ( n − x ) x + ( n − x ) x . By derivative, we know that f ( x ) is a strictlyincreasing function in the interval [1 , ⌊ n ⌋ ]. Thus E ( K , n − ) = f (1) ≤ E ( K s, t ) = f ( t ) ≤ f ( ⌊ n ⌋ ) = E ( K ⌈ n ⌉ , ⌊ n ⌋ ) . This completes the proof. ✷ Theorem 4.4 Let G be a graph with n vertices. Then ρ ≤ E ( G ) ≤ ρ + q ( n − F − ρ ) . Proof. Since n P i =1 ρ i = 0, we have E ( G ) = ρ + n P i =2 | ρ i | ≥ ρ + | n P i =2 ρ i | = 2 ρ . Since E ( G ) = ρ + n P i =2 | ρ i | , by the Cauchy-Schwartz inequality, we have E ( G ) ≤ ρ + q ( n − F − ρ ) . This completes the proof. ✷ Theorem 4.5 Let G be a non-trivial graph.Then E ( G ) ≥ s tr ( S ) tr ( S ) . Proof. Let a i = | ρ i | , b i = | ρ i | , p = and q = 3 in the H¨older inequality n X i =1 a i b i ≤ n X i =1 a pi ! p n X i =1 b qi ! q . Then n X i =1 | ρ i | = n X i =1 | ρ i | (cid:0) | ρ i | (cid:1) ≤ n X i =1 | ρ i | ! n X i =1 | ρ i | ! , E ( G ) ≥ n P i =1 | ρ i | (cid:18) n P i =1 | ρ i | (cid:19) = s tr ( S ) tr ( S ) . The proof is completed. ✷ For a graph G , we use M k ( G ) to denote the set of all k -matchings of G . If e = v i v j ∈ E ( G ), then we denote SO G ( e ) = SO G ( v i v j ) = (cid:16)q d i + d j (cid:17) = d i + d j , and we say that SO G ( e ) is the SO -value of the edge e . If β = { e , e , . . . , e k } ∈ M k ( G ), wecall that Q ki =1 SO G ( e i ) is the SO -value of matching β , and write SO G ( β ) = Q ki =1 SO G ( e i ).If G is a bipartite graph with n vertices, then the characteristic polynomial of G canbe written as (see [13, 22]) φ A ( G, x ) = | xI − A ( G ) | = ⌊ n ⌋ X k =0 ( − k m ( G, k ) x n − k , where m ( G, 0) = 1, and m ( G, k ) equals the number of k -matchings of G for 1 ≤ k ≤ ⌊ n ⌋ .The energy of G can be expressed as the Coulson integral formula (see [13, 22]) E A ( G ) = 1 π Z + ∞−∞ x ln ⌊ n ⌋ X k =1 m ( G, k ) x k dx. Then E A ( G ) is a strictly monotonously increasing function of m ( G, k ).For a bipartite graph G with n vertices, the adjacency matrix A ( G ) and Sombormatrix S ( G ) are nonnegative real symmetric matrices with zero diagonal, and A ( G ) and S ( G ) have the same zero-nonzero pattern, that is, for any 1 ≤ i, j ≤ n , ( i, j )-entry of A ( G ) is nonzero (or zero) if and only if ( i, j )-entry of S ( G ) is nonzero (or zero). Thus theSombor characteristic polynomial of G can be written as φ SO ( G, x ) = | xI − S ( G ) | = ⌊ n ⌋ X k =0 ( − k b ( S ( G ) , k ) x n − k , b ( S ( G ) , 0) = 1, and b ( S ( G ) , k ) equals the sum of SO -values of all k -matchings of G for 1 ≤ k ≤ ⌊ n ⌋ . Similarly, the Coulson integral formula for Sombor energy of a bipartitegraph G can be expressed as follows E ( G ) = 1 π Z + ∞−∞ x ln ⌊ n ⌋ X k =1 b ( S ( G ) , k ) x k dx. (4 . E ( G ) is a strictly monotonously increasing function of b ( S ( G ) , k ).So the following two results are direct. Proposition 4.6 Let G and G be two bipartite graphs with n vertices, and their Somborcharacteristic polynomials be φ SO ( G , x ) = ⌊ n ⌋ X k =0 ( − k b ( S ( G ) , k ) x n − k , φ SO ( G , x ) = ⌊ n ⌋ X k =0 ( − k b ( S ( G ) , k ) x n − k , respectively. If b ( S ( G ) , k ) ≥ b ( S ( G ) , k ) for all k ≥ , and there is a positive integer k such that b ( S ( G ) , k ) > b ( S ( G ) , k ) , then S ( G ) > S ( G ) . Proposition 4.7 Let G be a bipartite of with n vertices, and both B and B be two n × n nonnegative real symmetric matrices having the same zero-nonzero pattern with S ( G ) . If B > S ( G ) > B , then E B > E ( G ) > E B , where E B is the sum of the absolute values of all eigenvalues of B . Theorem 4.8 Let G be a bipartite graph with n vertices. Then E ( G − e ) < E ( G ) . Proof. Since S ( G − e ) < S ( G ) , by Proposition 4.7, we have the proof. ✷ The inverse sum indeg index, introduced by Vukiˇcevi´c and Gaˇsperov [29], is definedas ISI ( G ) = X v i v j ∈ E ( G ) d i d j d i + d j . Xu et al. [31] obtained lower and upper bounds on the inverse sum indeg energy E ISI ( G )of a graph G , and characterized the respective extremal graphs.12 heorem 4.9 Let G be a bipartite graph with n vertices. Then E ( G ) ≥ √ E ISI ( G ) with equality if and only if G is regular. Proof. Since q d i + d j ≥ √ d i d j d i + d j with equality if and only if G is regular, by Proposition4.7, we have the proof. ✷ Theorem 4.10 Let T n be a tree with n vertices and maximum degree ∆ . Then √ n − < E ( T n ) < √ 2∆ csc π n + 1) − √ , if n is even , √ 2∆ cot π n + 1) − √ , if n is odd . Proof. Since √ < q d i + d j ≤ √ 2∆ for a tree, by Proposition 4.7, we have √ E A ( T n ) < E ( T n ) < √ E A ( T n ). By Lemma 2.6, we have the proof. ✷ Problem 4.11 For a given class of graphs, characterize the graphs with the maximum orminimum Sombor energy. On the Sombor Estrada index Theorem 5.1 Let G be a graph with n vertices and m edges. Then EE ( G ) ≤ n − tr ( S )6 + tr ( S )24 + e √ F − √ F − F √ F − F . Equality holds if and only if G is an empty graph. Proof. By the definition of the Sombor Estrada index of graphs, we have EE ( G ) ≤ n + n X t =1 ρ t ( G ) + n X t =1 ρ t ( G )2! + n X t =1 ρ t ( G )3! + n X t =1 ρ t ( G )4! + n X t =1 ∞ X k ≥ | ρ t ( G ) | k k != n + F + tr ( S )6 + tr ( S )24 + n X t =1 ∞ X k ≥ | ρ t ( G ) | k k != n + F + tr ( S )6 + tr ( S )24 + ∞ X k ≥ k ! n X t =1 | ρ t ( G ) | k n + F + tr ( S )6 + tr ( S )24 + ∞ X k ≥ k ! n X t =1 ( ρ t ( G )) k ≤ n + F + tr ( S )6 + tr ( S )24 + ∞ X k ≥ k ! ( n X t =1 ρ t ( G )) k = n + F + tr ( S )6 + tr ( S )24 + ∞ X k =0 (2 F ) k k ! − − √ F − F − F √ F − F = n − tr ( S )6 + tr ( S )24 + e √ F − √ F − F √ F − F . This completes the proof. ✷ Theorem 5.2 Let G be a graph with n vertices. Then EE ( G ) ≥ e ρ ( G ) + n + ( p − e E ( G )2 − ρ G ) p − + qe − E ( G )2 q (5 . with equality if and only if ρ ( G ) = · · · = ρ p ( G ) and ρ n − q +1 ( G ) = · · · = ρ n ( G ) , where p , n and q are the number of positive, zero and negative Sombor eigenvalues of G , respectively. Proof. Let ρ ( G ) ≥ ρ ( G ) ≥ · · · ≥ ρ p ( G ) be the positive, and ρ n − q +1 ( G ) ≥ ρ n − q +2 ( G ) ≥· · · ≥ ρ n ( G ) be the negative Sombor eigenvalues of G . By the arithmetic-geometric meaninequality, we have p X i =2 e ρ i ( G ) ≥ ( p − e ρ G )+ ··· + ρp ( G ) p − = ( p − e E ( G )2 − ρ G ) p − . (5 . n X i = n − q +1 e ρ i ( G ) ≥ qe − E ( G )2 q . (5 . n − q X p +1 e ρ i ( G ) = n . Thus we have EE ( G ) ≥ e ρ ( G ) + n + ( p − e E ( G )2 − ρ G ) p − + qe − E ( G )2 q . The equality holds in (5 . 1) if and only if equality holds in both (5 . 2) and (5 . 3) andthese happen if and only if ρ ( G ) = · · · = ρ p ( G ) and ρ n − q +1 ( G ) = · · · = ρ n ( G ). Thiscompletes the proof. ✷ heorem 5.3 Let G be a bipartite graph with n vertices. Then EE ( G ) ≥ n + 2 cosh( ρ ( G )) + ( r − 2) cosh (cid:18) E ( G ) − ρ ( G ) r − (cid:19) (5 . with equality if and only if ρ ( G ) = · · · = ρ p ( G ) , where r is the rank of Sombor matrix. Proof. Since G is bipartite, we have that its Sombor eigenvalues are symmetric withrespect to zero, i.e. ρ i ( G ) = − ρ n − i +1 ( G ) for i = 1 , , . . . , ⌊ n ⌋ . By a similar argument asthe proof of Theorem 5.2, we have EE ( G ) = n + e ρ ( G ) + e − ρ ( G ) + p X i =2 e ρ i ( G ) + p X i =2 e − ρ i ( G ) ≥ n + e ρ ( G ) + e − ρ ( G ) +( p − (cid:18) e E ( G )2 − ρ G ) p − + e − E ( G )2 − ρ G ) p − (cid:19) = n + 2 cosh( ρ ( G )) + ( r − 2) cosh (cid:18) E ( G ) − ρ ( G ) r − (cid:19) . Note that r = 2 p . Equality holds in (5 . 4) if and only if ρ ( G ) = · · · = ρ p ( G ). The proofis completed. ✷ Theorem 5.4 Let G be a connected bipartite graph with n ≥ vertices. Then EE ( G ) ≤ n − √ F (5 . with equality if and only if G ∼ = K s, t , s + t = n . Proof. Since G is bipartite, we have that its Sombor eigenvalues are symmetric withrespect to zero. Let p and n be the number of positive and zero Sombor eigenvalues of G , respectively. Then we have EE ( G ) = n + p X i =1 (cid:0) e ρ i ( G ) + e − ρ i ( G ) (cid:1) = n + 2 p + 2 ∞ X k =1 p P i =1 ρ ki ( G )(2 k )! ≤ n + 2 ∞ X k =1 (cid:18) p P i =1 ρ i ( G ) (cid:19) k (2 k )!15 n − ∞ X k =0 ( √ F ) k (2 k )!= n − e √ F + e −√ F = n − √ F . If equality holds in (5.5), then p X i =1 ρ ki ( G ) = p X i =1 ρ i ( G ) ! k for k ≥ 1. Since G is a connected graph with n ≥ ρ ( G ) > 0, that is p ≥ 1. For k ≥ p X i =1 ρ ki ( G ) = p X i =1 ρ i ( G ) ! k implies that p ≤ 1, as ρ i ( G )’s are positive eigenvalues. Thus p = 1, that is ρ ( G ) = − ρ n ( G ) = √ F , ρ ( G ) = · · · = ρ n − ( G ) = 0. By Lemma 2.7, we have diam ( G ) = 2. Thus G is a complete bipartite graph K s, t , s + t = n . Conversely, if G ∼ = K s, t , then S ( G ) = √ s + t A ( G ). Thus ρ ( G ) = − ρ n ( G ) = √ s t + st = √ F , ρ ( G ) = · · · = ρ n − ( G ) = 0.It is easy to check that equality holds in (5.5). The proof is completed. ✷ Corollary 5.5 Let G be a complete bipartite graph with n ≥ vertices. Then n − p ( n − n − n + 2) ≤ EE ( G ) ≤ n − rl n m j n k + j n k l n m The equality in the left hand side holds if and only if G ∼ = K , n − , and the equality in theright hand side holds if and only if G ∼ = K ⌈ n ⌉ , ⌊ n ⌋ . Proof. By the proof of Theorem 5.4, we have EE ( K s, t ) = n − √ s t + st = n − p ( n − t ) t + ( n − t ) t , ≤ t ≤ j n k . Let f ( x ) = n − p ( n − x ) x + ( n − x ) x . By derivative, we know that f ( x ) isa strictly increasing function in the interval [1 , ⌊ n ⌋ ]. Thus EE ( K , n − ) = f (1) ≤ EE ( K s, t ) = f ( t ) ≤ f ( ⌊ n ⌋ ) = EE ( K ⌈ n ⌉ , ⌊ n ⌋ ) . This completes the proof. ✷ eferences [1] L. Collatz, U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Sem. Univ. Hamburg 21(1957) 63-77.[2] R. Cruz, I. Gutman, J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399(2021) 126018.[3] K.Ch. Das, A.S. C¸ evik, I.N. Cangul, Y. Shang, On Sombor Index, Symmetry, 13, (2021)140.[4] H. Deng, Z. Tang, R. Wu, Molecular trees with extremal values of Sombor indices, Int JQuantum Chem. DOI: 10.1002/qua.26622.[5] B. Furtula, I. Gutman, A forgotten topological index, J Math Chem , 53 (2015) 1184-1190.[6] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz 103 (1978)1-22.[7] I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total π -electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456.[8] I. Gutman, Comparative studies of graph energies, Bull. Acad. Serbe Sci. Arts (Cl. Sci.Math. Natur.) 144 (2012) 1-17.[9] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R.Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin,2001, pp. 196-211.[10] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices,MATCH Commun. Math. Comput. Chem. 86 (2021) 11-16.[11] I. Gutman, Some basic properties of Sombor indices, Open J. Discret. Appl. Math. 4 (2021)1-3.[12] C. Godsil, Algebraic Combinatorics, CRC Press, Boca Raton, 1993.[13] I. Gutman, O.E. Polansky, Mathatical Concepts in Organic Chemistry, Springer, Berlin,1986.[14] I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. Total π -electron energy ofalternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.[15] W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995)593-616.[16] Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl. 108 (1988) 135-140.[17] X. Li, Z. Wang, Trees with extremal spectral radius of weighted adjacencymatrices among trees weighted by degree-based indices, Linear Algebra Appl.https://doi.org/10.1016/j.laa.2021.02.023. 18] V.R. Kulli, Sombor indices of certain graph operators, International Journal of EngineeringSciences & Research Technology, 10 (2021) 127-134.[19] H. Kober, On the arithmetic and geometric means and on H¨olders inequality, Proc. Amer.Math. Soc. 9 (1958) 452-459.[20] V.R. Kulli, I. Gutman, Computation of Sombor Indices of Certain Networks, InternationalJournal of Applied Chemistry, 8 (2021) 1-5.[21] R. Liu, W.C. Shiu, General Randi´c matrix and general Randi´c incidence matrix, DiscreteAppl. Math. 186 (2015) 168-175.[22] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012.[23] I. Milovanovi´c, E. Milovanovi´c, M. Mateji´c, On Some mathematical properties of Somborindeces, Bull. Int. Math. Virtual Inst. 11 (2021) 341-353.[24] V. Nikiforov, Some new results in extremal graph theory. Surveys in combinatorics 2011,141-181, London Math. Soc. Lecture Note Ser., 392, Cambridge Univ. Press, Cambridge,2011.[25] I. Redˇzepovi´c, Chemical applicability of Sombor indices, J. Serb. Chem. Soc.https://doi.org/10.2298/JSC201215006R.[26] T. R´eti, T. Doˇsli´c, A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11-18.[27] D. Stevanovi´c, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.[28] M. Taita, J. Tobin, Three conjectures in extremal spectral graph theory, J. Comb. Theory,Ser. B 126 (2017) 137-161.[29] D. Vukiˇcevi´c, M. Gaˇsperov, Bond additive modelling 1. Adriatic indices, Croat. Chem. Acta83 (2010) 261-273.[30] Z. Wang, Y. Mao, Y. Li, B. Furtula, On relations between Sombor and other degree-basedindices, J. Appl. Math. Comput. https://doi.org/10.1007/s12190-021-01516-x.[31] B. Xu, S. Li, R. Yu, Q. Zhao, On the spectral radius and energy of the weighted adjacencymatrix of a graph, Appl. Math. Comput. 340 (2019) 156-163.[32] B. Zhou, On the spectral radius of nonnegative matrices, Australas. J. Comb. 22 (2000)301-306.18] V.R. Kulli, Sombor indices of certain graph operators, International Journal of EngineeringSciences & Research Technology, 10 (2021) 127-134.[19] H. Kober, On the arithmetic and geometric means and on H¨olders inequality, Proc. Amer.Math. Soc. 9 (1958) 452-459.[20] V.R. Kulli, I. Gutman, Computation of Sombor Indices of Certain Networks, InternationalJournal of Applied Chemistry, 8 (2021) 1-5.[21] R. Liu, W.C. Shiu, General Randi´c matrix and general Randi´c incidence matrix, DiscreteAppl. Math. 186 (2015) 168-175.[22] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012.[23] I. Milovanovi´c, E. Milovanovi´c, M. Mateji´c, On Some mathematical properties of Somborindeces, Bull. Int. Math. Virtual Inst. 11 (2021) 341-353.[24] V. Nikiforov, Some new results in extremal graph theory. Surveys in combinatorics 2011,141-181, London Math. Soc. Lecture Note Ser., 392, Cambridge Univ. Press, Cambridge,2011.[25] I. Redˇzepovi´c, Chemical applicability of Sombor indices, J. Serb. Chem. Soc.https://doi.org/10.2298/JSC201215006R.[26] T. R´eti, T. Doˇsli´c, A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11-18.[27] D. Stevanovi´c, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.[28] M. Taita, J. Tobin, Three conjectures in extremal spectral graph theory, J. Comb. Theory,Ser. B 126 (2017) 137-161.[29] D. Vukiˇcevi´c, M. Gaˇsperov, Bond additive modelling 1. Adriatic indices, Croat. Chem. Acta83 (2010) 261-273.[30] Z. Wang, Y. Mao, Y. Li, B. Furtula, On relations between Sombor and other degree-basedindices, J. Appl. Math. Comput. https://doi.org/10.1007/s12190-021-01516-x.[31] B. Xu, S. Li, R. Yu, Q. Zhao, On the spectral radius and energy of the weighted adjacencymatrix of a graph, Appl. Math. Comput. 340 (2019) 156-163.[32] B. Zhou, On the spectral radius of nonnegative matrices, Australas. J. Comb. 22 (2000)301-306.