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Abstract In a network or a graph, the total communicability (TC) has been defined as the sum of the entries in the exponential of the adjacency matrix of the network. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. In this paper, we obtain some bounds for total communicability of a graph G , T C (G) , in terms of spectral radius of the adjacency matrix, number of vertices, number of edges, minimum degree and the maximum degree of G. Moreover, we find some upper bounds for T C (G) when G is the Cartesian product, tensor product or the strong product of two graphs. In addition, Nordhaus–Gaddum-type results for the total communicability of a graph G are established. [ABSTRACT FROM AUTHOR]

Abstract Let G = (V , E) be a graph with vertex set V and edge set E. The v e -degree of a vertex v ∈ V equals the number of edges v e -dominated by v and the e v -degree of an edge e ∈ E equals the number of vertices e v -dominated by e. Recently, Chellali et al. studied the properties of v e -degree and e v -degree of graphs (Chellali et al., 2017). Also they focused on the regularity and irregularity of these types of degrees and proposed several open problems. In this paper, we solve one of them and obtain some results on the regularity and irregularity of v e - and e v -degrees in graphs. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph of order n and size m , with vertex set V ( G ) = { v 1 , v 2 , … , v n } , without isolated vertices and sequence of vertex degrees Δ = d 1 ≥ d 2 ≥ ⋯ ≥ d n = δ > 0 , d i = d G ( v i ) . If the vertices v i and v j are adjacent, we denote it as v i v j ∈ E ( G ) or i ∼ j . With TI we denote a topological index that can be represented as T I = T I ( G ) = ∑ i ∼ j F ( d i , d j ) , where F is an appropriately chosen function with the property F ( x , y ) = F ( y , x ) . A general extended adjacency matrix A = ( a i j ) of G is defined as a i j = F ( d i , d j ) if the vertices v i and v j are adjacent, and a i j = 0 otherwise. Denote by f i , i = 1 , 2 , … , n the eigenvalues of A . The “energy” of the general extended adjacency matrix is defined as E T I = E T I ( G ) = ∑ i = 1 n | f i | . Lower and upper bounds on E T I are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n . The energy E ( G ) of a simple graph G is the sum of absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy, the signless Laplacian energy and the distance energy of graph G are denoted by L E ( G ) , S L E ( G ) and D E ( G ) , respectively. In this paper we introduce a distance Laplacian energy D L E and distance signless Laplacian energy D S L E of a connected graph. We present Nordhaus–Gaddum type bounds on Laplacian energy L E ( G ) and signless Laplacian energy S L E ( G ) in terms of order n of graph G and characterize graphs for which these bounds are best possible. The complete graph and the star give the smallest distance signless Laplacian energy D S L E among all the graphs and trees of order n , respectively. We give lower bounds on distance Laplacian energy D L E in terms of n for graphs and trees, and characterize the extremal graphs. Also we obtain some relations between D E , D S L E and D L E of graph G . Moreover, we give several open problems in this paper. [ABSTRACT FROM AUTHOR]

Let d ( u, v ) be the distance between u and v of graph G , and let W f ( G ) be the sum of f ( d ( u, v )) over all unordered pairs { u, v } of vertices of G , where f ( x ) is a function of x . In some literatures, W f ( G ) is also called the Q -index of G . In this paper, some unified properties to Q -indices are given, and the majorization theorem is illustrated to be a good tool to deal with the ordering problem of Q -index among trees with n vertices. With the application of our new results, we determine the four largest and three smallest (resp. four smallest and three largest) Q -indices of trees with n vertices for strictly decreasing (resp. increasing) nonnegative function f ( x ), and we also identify the twelve largest (resp. eighteen smallest) Harary indices of trees of order n  ≥ 22 (resp. n  ≥ 38) and the ten smallest hyper-Wiener indices of trees of order n  ≥ 18, which improve the corresponding main results of Xu (2012) and Liu and Liu (2010), respectively. Furthermore, we obtain some new relations involving Wiener index, hyper-Wiener index and Harary index, which gives partial answers to some problems raised in Xu (2012). [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix and the signless Laplacian matrix of G are L(G)=D(G)-A(G) and Q(G)=D(G)+A(G), respectively. Also denote by λ1(G), a(G), q1(G) and δ(G) the largest eigenvalue of A(G), the second smallest eigenvalue of L(G), the largest eigenvalue of Q(G) and the minimum degree of G, respectively. In this paper, we give partial proofs to the following two conjectures: (i)Aouchiche (Comparaison Automatisée d’Invariants en Théorie des Graphes, 2006) if G is a connected graph, then a(G)/δ(G) is minimum for graph composed of 2 triangles linked with a path. (ii)Aouchiche et al. (Linear Algebra Appl 432:2293-2322, 2010) and Cvetković et al. (Publ Inst Math Beogr 81(95):11-27, 2007) if G is a connected graph with n≥4 vertices, then q1(G)-2λ1(G)≤n-2n-1 with equality holding if and only if G≅K1,n-1. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a connected graph with vertex set V(G) = {v1, v2,..., vn} and edge set E(G). Let D(G) be the distance matrix of G and λ1(D) ≥···≥ λn(D) be its distance spectrum. The distance between distance spectra of G and G' is defined by σ(G,G') = i=1∑n|λi(D(G)) - λi(D(G'))|. Define the cospectrality of G by cs(G) = min {i=1∑n|λi(D(G)) - λi(D(G'))|: G' not isomorphic to G}. Let csn = max{cs(G) : G a connected graph on n vertices}. In the paper, we obtain lower bounds on σ(G, Kn) and σ(G, Ka,b) for a+b = n. Furthermore, we give an upper bound on csn. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph of order n . The normalized Laplacian eigenvalues of graph G are denoted by ρ 1 ( G ) ≥ ρ 2 ( G ) ≥ ⋯ ≥ ρ n − 1 ( G ) ≥ ρ n ( G ) = 0 . Also let G and G ′ be two nonisomorphic graphs on n vertices. Define the distance between the normalized Laplacian spectra of G and G ′ as σ N ( G , G ′ ) = ∑ i = 1 n | ρ i ( G ) − ρ i ( G ′ ) | p , p ≥ 1 . Define the cospectrality of G by c s N ( G ) = min ⁡ { σ N ( G , G ′ ) : G ′ not isomorphic to G } . Let c s n N = max ⁡ { c s N ( G ) : G a graph on n vertices } . In this paper, we give an upper bound on c s N ( G ) in terms of the graph parameters. Moreover, we obtain an exact value of c s n N . An upper bound on the distance between the normalized Laplacian spectra of two graphs has been presented in terms of Randić energy. As an application, we determine the class of graphs, which are lying closer to the complete bipartite graph than to the complete graph regarding the distance of normalized Laplacian spectra. [ABSTRACT FROM AUTHOR]

The aim of this note is to establish new spectral bounds for the harmonic matrix. [ABSTRACT FROM AUTHOR]

Letdenote the class ofc-cyclic graphs withnvertices, girthandpendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connectedc-cyclic graphs withvertices, girthgand at mostpendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius ofwhenand, and we also identify the unique extremal graph with largest Laplacian spectral radius inin the caseand eitherandgis even orandgis odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391]. [ABSTRACT FROM AUTHOR]

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A kite graph Ki n ,  ω is a graph obtained from a clique K ω and a path P n − ω by adding an edge between a vertex from the clique and an endpoint from the path. In this note, we prove that K i n , n − 1 is determined by its signless Laplacian spectrum when n ≠ 5 and n ≥ 4, and K i n , n − 1 is also determined by its distance spectrum when n ≥ 4. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n . For i = 1 , 2 , … , n , let d i be the degree of the vertex v i of G . The extended adjacency matrix A ex of G is defined so that its ( i, j )-entry is equal to 1 2 ( d i d j + d j d i ) if the vertices v i and v j are adjacent, and 0 otherwise,Yang et al. (1994). The spectral radius η 1 and the energy E e x of the A ex -matrix are examined. Lower and upper bounds on η 1 and E e x are obtained, and the respective extremal graphs characterized. [ABSTRACT FROM AUTHOR]

We study the Laplacian and the signless Laplacian energy of connected unicyclic graphs, obtaining a tight upper bound for this class of graphs. We also find the connected unicyclic graph on n vertices having largest (signless) Laplacian energy for 3 ≤ n ≤ 13 . For n ≥ 11 , we conjecture that the graph consisting of a triangle together with n − 3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on n vertices. We prove this conjecture for many classes of graphs, depending on σ , the number of (signless) Laplacian eigenvalues bigger than or equal to 2. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ . Let ( d ) = ( d 1 , d 2 , … , d n ) and ( d ⁎ ) = ( d 1 ⁎ , d 2 ⁎ , … , d n ⁎ ) be the sequences of degrees and conjugate degrees of G . We define π = ∑ i = 1 n d i and π ⁎ = ∑ i = 1 n d i ⁎ , and prove that π ⁎ ≤ L E L ≤ I E ≤ π where LEL and IE are, respectively, the Laplacian-energy-like invariant and the incidence energy of G . Moreover, we prove that π − π ⁎ > ( δ / 2 ) ( n − Δ ) for a certain class of graphs. Finally, we compare the energy of G and π , and present an upper bound for the Laplacian energy in terms of degree sequence. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n with diameter d . Remoteness ρ of G is the maximum average distance from a vertex to all others and ∂ 1 ≥ ⋯ ≥ ∂ n are the distance eigenvalues of G . Aouchiche and Hansen (0000), Aouchiche and Hansen conjectured that ρ + ∂ 3 > 0 when d ≥ 3 and ρ + ∂ ⌊ 7 d 8 ⌋ > 0 . In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ∂ n + ρ and ∂ 1 − ρ when G ≇ K n and the extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

The classical first and second Zagreb indices of a graph G are defined as M 1 ( G ) = ∑ v ∈ V d G ( v ) 2 and M 2 ( G ) = ∑ u v ∈ E ( G ) d G ( u ) d G ( v ) , where d G ( v ) is the degree of the vertex v of graph G . Recently, Furtula et al. (2014) studied the difference between the Zagreb indices and mentioned a problem to characterize the graphs for which M 1 ( G ) > M 2 ( G ) or M 1 ( G ) < M 2 ( G ) or M 1 ( G ) = M 2 ( G ) . In this paper we completely solve this problem. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n with m edges and clique number ω . Let μ 1 ≥ μ 2 ≥ … ≥ μ n = 0 be the Laplacian eigenvalues of G and let σ = σ ( G ) ( 1 ≤ σ ≤ n ) be the largest positive integer such that μ σ ≥ 2 m n . In this paper we study the relation between σ and ω . In particular, we provide the answer to Problem 2.3 raised in Pirzada and Ganie (2015) [15] . Moreover, we characterize all connected threshold graphs with σ < ω − 1 , σ = ω − 1 and σ > ω − 1 . We obtain Nordhaus–Gaddum-type results for σ . Some relations between σ with other graph invariants are obtained. [ABSTRACT FROM AUTHOR]

The Kirchhoff indexof a graphGis the sum of resistance distances between all unordered pairs of vertices, which was introduced by Klein and Randić. In this paper, we characterize all extremal graphs with respect to Kirchhoff index among all graphs obtained by deletingpedges from a complete graphwithand obtain a sharp upper bound on the Kirchhoff index of these graphs. In addition, all the graphs with the first to ninth maximal Kirchhoff indices are completely determined among all connected graphs of order. [ABSTRACT FROM PUBLISHER]

Let G = ( V , E ) be a connected graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The transmission Tr ( v i ) of vertex v i is defined to be the sum of distances from v i to all other vertices. Let Tr ( G ) be the n × n diagonal matrix with its ( i , i ) -entry equal to Tr G ( v i ) . The distance signless Laplacian is defined as D Q ( G ) = Tr ( G ) + D ( G ) , where D ( G ) is the distance matrix of G . Let ∂ 1 ( G ) ≥ ∂ 2 ( G ) ≥ ⋯ ≥ ∂ n ( G ) denote the eigenvalues of distance signless Laplacian matrix of G . In this paper, we first characterize all graphs with ∂ n ( G ) = n − 2 . Secondly, we characterize all graphs with ∂ 2 ( G ) ∈ [ n − 2 , n ] when n ≥ 11 . Furthermore, we give the lower bound on ∂ 2 ( G ) with independence number α and the extremal graph is also characterized. [ABSTRACT FROM AUTHOR]

A complete split graph C S ( n , α ) , is a graph on n vertices consisting of a clique on n − α vertices and an independent set on the remaining α ( 1 ≤ α ≤ n − 1 ) vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, we prove that C S ( n , α ) is determined by its Laplacian spectrum when 1 ≤ α ≤ n − 1 , and C S ( n , α ) is also determined by its signless Laplacian spectrum when 1 ≤ α ≤ n − 1 and α ≠ 3 . [ABSTRACT FROM AUTHOR]

The Wiener polarity index W p ( G ) of a graph G is the number of unordered pairs of vertices { u ,  v } in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for W p ( G ). Our lower bound on W p ( G ) + W p ( G ¯ ) is always better than the previous lower bound given by Zhang and Hu. [ABSTRACT FROM AUTHOR]

The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

The Graovac–Ghorbani index ( ABC GG ), was introduced by Graovac and Ghorbani, (2010). In this paper we present some upper and lower bounds on the ABC GG index and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

The zeroth-order Randić index and the sum-connectivity index are very popular topological indices in mathematical chemistry. These two indices are based on vertex degrees of graphs and attracted a lot of attention in recent years. Recently Li and Li (2015) studied these two indices for trees of order n . In this paper we obtain a relation between the zeroth-order Randić index and the sum-connectivity index for graphs. From this we infer an upper bound for the sum-connectivity index of graphs. Moreover, we prove that the zeroth-order Randić index is greater than the sum-connectivity index for trees. Finally, we show that R 2,  α ( G ) is greater or equal R 1,  α ( G ) ( α ≥ 1) for any graph and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected threshold graph of order n with m edges and trace T . In this paper we give a lower bound on Laplacian energy in terms of n, m and T of G . From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies. [ABSTRACT FROM AUTHOR]

Let G be a bipartite graph of order n with m edges. The energy E ( G ) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A . In 1974, one of the present authors established lower and upper bounds for E ( G ) in terms of n, m , and det A . Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m , and the first Zagreb index is obtained, and the extremal graphs characterized. [ABSTRACT FROM AUTHOR]

The connective eccentricity index (CEI) of a graph G is defined as ξ ce ( G ) = ∑ v i ∈ V ( G ) d ( v i ) ε ( v i ) where ε ( v i ) and d ( v i ) are the eccentricity and the degree of vertex v i , respectively, in G . In this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number β and characterize the corresponding extremal trees. And the maximal graphs of order n and with matching number β and n edges have been determined which maximize the connective eccentricity index. Also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number β . Moreover we establish some relations between connective eccentricity index and eccentric connectivity index, as another eccentricity-based invariant, of graphs. [ABSTRACT FROM AUTHOR]

Let W, Sz, PI , and WP be, respectively, the Wiener, Szeged, PI, and Wiener polarity indices of a molecular graph G . Let M 1 and M 2 be the first and second Zagreb indices of G . We obtain relations between these classical distance– and degree–based topological indices. [ABSTRACT FROM AUTHOR]

The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which the bound is attained. Moreover, we obtain some relations between the Zagreb indices for unicyclic graphs. [ABSTRACT FROM AUTHOR]

For any real number α , let s α ( G ) denote the sum of the α th power of the non-zero Laplacian eigenvalues of a graph G . In this paper, we first obtain sharp bounds on the largest and the second smallest Laplacian eigenvalues of a graph, and a new spectral characterization of a graph from its Laplacian eigenvalues. Using these results, we then establish sharp bounds for s α ( G ) in terms of the number of vertices, number of edges, maximum vertex degree and minimum vertex degree of the graph G , from which a Nordhaus–Gaddum type result for s α is also deduced. Moreover, we characterize the graphs maximizing s α for α > 1 among all the connected graphs with given matching number. [ABSTRACT FROM AUTHOR]

Let S z ( G ) and ξ c ( G ) be the Szeged index and the eccentric connectivity index of a graph G , respectively. In this paper we obtain a lower bound on S z ( T ) − ξ c ( T ) by double counting on some matrix and characterize the extremal graphs. From this result we compare the Szeged index and the eccentricity connectivity index of trees. For bipartite graphs we also compare the Szeged index and the eccentricity connectivity index. Moreover, we show that S z ( G ) − ξ c ( G ) ≥ − 4 for bipartite graphs and this result is not true in the general case. Finally, we classify the bipartite graphs G in which S z ( G ) − ξ c ( G ) ∈ { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 } . [ABSTRACT FROM AUTHOR]

Resistance distance was introduced by Klein and Randić as a generalisation of the classical distance. The Kirchhoff index Kf(G) of a graph G is the sum of resistance distances between all unordered pairs of vertices. In this article we characterise the extremal graphs with the maximal Kirchhoff index among all non-trivial quasi-tree graphs of order n. Moreover, we obtain a lower bound on the Kirchhoff index for all non-trivial quasi-tree graphs of order n. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a simple graph with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) . The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G , defined as Q ( G ) = Tr ( G ) + D ( G ) , where Tr ( G ) is the diagonal matrix of vertex transmissions of G and D ( G ) is the distance matrix of G . In [10] , Xing et al. determined the graph with minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. For n ≥ 3 , let T n − 3 , 1 1 be the n -vertex tree of maximum degree n − 2 . In this paper, we show that T n − 3 , 1 1 gives the second minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. Moreover, we prove two conjectures involving the second largest eigenvalue of the distance signless Laplacian matrix Q ( G ) of graph G . [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V = { v 1 , v 2 , … , v n } . Also let μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n − 1 ( G ) ≥ μ n ( G ) = 0 and q 1 ( G ) ≥ q 2 ( G ) ≥ ⋯ ≥ q n ( G ) ≥ 0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G , respectively. In this paper we obtain μ i ( G ) ≤ i − 1 + min U i ⁡ max ⁡ { | N H ( v k ) ∪ N H ( v j ) | : v k v j ∈ E ( H ) } , where N H ( v k ) is the set of neighbors of vertex v k in V ( H ) = V ( G ) \ U i , U i is any ( i − 1 ) -subset of V ( G ) (here, we agree that i ∈ { 1 , … , n − 1 } and μ i ( G ) ≤ i − 1 if E ( H ) = ∅ ). For any graph G , this bound does not exceed the order of G . Moreover, we prove that max ⁡ { μ i ( G ) , q i ( G ) } ≤ max i ≤ k ≤ n ⁡ { d G ( v k ) + ∑ v j ∈ N G ( v k ) ∩ N d G ( v j ) d G ( v k ) } ≤ 2 d G ( v i ) , where d G ( v i ) is the i -th largest degree of G and N = { v i , v i + 1 , … , v n } . [ABSTRACT FROM AUTHOR]

Abstract: Molecular descriptors play an important role in mathematical chemistry, especially in the QSPR and QSAR modeling. Among them, a special place is reserved for the so called topological indices. Nowadays, there exists a legion of topological indices that found applications in various areas of chemistry Todeschini and Consonni (2000, 2009). Recently, we carried out comparison between several topological indices for various classes of graphs and trees (Das et al., 2012; Das and Trinajsti, 2010, 2011, 2012; Horoldagva and Das, 2012; Hua and Das, 2013). In this report, we compare the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and edges. Also let be the eigenvalues of the Laplacian matrix of graph . The Laplacian energy of the graph is defined as In this paper, we present some lower and upper bounds for of graph in terms of , the number of edges and the maximum degree . Also we give a Nordhaus–Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs. [Copyright &y& Elsevier]

Abstract: In Das et al. (2013) [8], a new graph on monogenic semigroups (with zero) having elements has been recently defined. The vertices are the non-zero elements and, for , any two distinct vertices and are adjacent if in . As a continuing study, in Akgunes et al. (2014) [3], it has been investigated some well known indices (first Zagreb index, second Zagreb index, Randić index, geometric–arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over . In the light of above references, our main aim in this paper is to extend these studies over to the tensor product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the tensor product of any two (not necessarily different) graphs and . [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The incidence matrix of G is the matrix whose -entry is 1 if is incident to and 0 otherwise. The incidence energy IE of G is the sum of the singular values of . In this paper we give lower and upper bounds for IE in terms of n, m, maximum degree, clique number, independence number, and the first Zagreb index. Moreover, we obtain Nordhaus–Gaddum-type results for IE. [Copyright &y& Elsevier]

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues . The Laplacian-energy-like invariant of the graph G is defined as Lower and upper bounds for LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The Laplacian matrix of G is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Let be the Laplacian eigenvalues of G. For a graph G and a real number , the graph invariant is the sum of the β-th power of the non-zero Laplacian eigenvalues of G, that is, In this paper, we obtain some lower and upper bounds on for G in terms of n, the number of edges m, maximum degree , clique number ω, independence number α and the number of spanning trees t. Moreover, we present some Nordhaus–Gaddum-type results for of G. [Copyright &y& Elsevier]

Abstract: Let be a simple connected graph with vertex set and edge set . The first Zagreb index and the second Zagreb index are defined as follows: and where is the degree of vertex in . The eccentric connectivity index of a graph , denoted by , is defined as where is the eccentricity of in . Recently, Das and Trinajstić (2011) [11] compared the eccentric connectivity index and Zagreb indices for chemical trees and molecular graphs. However, the comparison between the eccentric connectivity index and Zagreb indices, in the case of general trees and general graphs, is very hard and remains unsolved till now. In this paper, we compare the eccentric connectivity index and Zagreb indices for some graph families. We first give some sufficient conditions for a graph satisfying , . Then we introduce two classes of composite graphs, each of which has larger eccentric connectivity index than the first Zagreb index, if the original graph has larger eccentric connectivity index than the first Zagreb index. As a consequence, we can construct infinite classes of graphs having larger eccentric connectivity index than the first Zagreb index. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The Laplacian matrix of G is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Among all eigenvalues of the Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic connectivity of a graph [18]. In this paper we give a lower bound on the algebraic connectivity of graphs. Moreover, we mention two conjectures, obtained by the system AutoGraphiX, about the algebraic connectivity , diameter and the minimum degree of graphs (see, [2], available online at http://www.gerad.ca/∼agx/):with equality if and only if G is isomorphic to a graph with and , and (ii) is minimum for a graph composed of 2 cliques on vertices with a common vertex if n is odd, and linked by an edge if n is even. Here we prove conjecture in (i) using the lower bound on the algebraic connectivity of graphs and conjecture in (ii), respectively. [Copyright &y& Elsevier]

Abstract: The power of a graph , denoted by , is the graph with the same vertex set as , such that two vertices are adjacent in if and only if their distance is at most in . In this paper, we give bounds on the first two largest Laplacian eigenvalues of the second power of a general graph, and on the second power of a tree. We also give a Nordhaus–Gaddum-type inequality for the Laplacian spectral radius of . [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]

The concept of atom-bond connectivity (ABC) index was introduced in the chemical graph theory in 1998. The atom-bond connectivity (ABC) index of a graph G defined as ABC(G) = ... √di + dj - 2/didj where E(G) is the edge set and d is the degree of vertex vi of G. Very recently Graovac et al. define a new version of the ABC index as ABC2 = ... √ni + j - 2/ninj where ni denotes the number of vertices of G whose distances to vertex vi are smaller than those to the other vertex vj of the edge e = vivj, and nj is defined analogously. In this paper we determine the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2). [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues . The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as and , respectively. We compare Kf and LEL and establish two sufficient conditions under which . The connected graphs of order n with nine greatest Kirchhoff indices are determined; for these holds. [Copyright &y& Elsevier]

Abstract: Let be a simple graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of is . In , Cvetković et al. (2007) have given conjectures on signless Laplacian eigenvalues of (see also Aouchiche and Hansen (2010) , Oliveira et al. (2010) ). Here we prove two conjectures. [Copyright &y& Elsevier]

Abstract: Let be a simple, connected and undirected graph with vertex set and edge set . Also let be the distance matrix of a graph (Janežič et al., 2007) . Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph. A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009)  and Papendieck and Recht (2000) . Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002)  and Das (2009) , respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. [Copyright &y& Elsevier]