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Abstract Let G = (V , E) be a simple graph of order n with normalized Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0. The normalized Laplacian spread of graph G , denoted by ρ 1 − ρ n − 1 , is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ 2 of graphs. Moreover, we give a lower bound on ρ 2 of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs. [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]

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]

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 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]

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]

For G being a graph with n vertices and m edges, and with Laplacian eigenvalues μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n − 1 ≥ μ n = 0 , the Laplacian energy is defined as L E = ∑ i = 1 n | μ i − 2 m / n | . Let σ be the largest positive integer such that μ σ ≥ 2 m / n . We characterize the graphs satisfying σ = n − 1 . Using this, we obtain lower bounds for LE in terms of n, m , and the first Zagreb index. In addition, we present some upper bounds for LE in terms of graph invariants such as n, m , maximum degree, vertex cover number, and spanning tree packing number. [ABSTRACT FROM AUTHOR]

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 connected graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . In this paper, we obtain some new and improved sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph G. [Copyright &y& Elsevier]

Abstract: We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained. [Copyright &y& Elsevier]

Abstract: Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on are given. We show that some known bounds can be obtained from our bounds. [Copyright &y& Elsevier]

Let G be a connected graph with vertex set V ( G ). For u, v ∈ V ( G ), d ( v ) and d ( u, v ) denote the degree of the vertex v and the distance between the vertices u and v . A much studied degree–and–distance–based graph invariant is the degree distance, defined as D D = ∑ { u , v } ⊆ V ( G ) [ d ( u ) + d ( v ) ] d ( u , v ) . A related such invariant (usually called “Gutman index”) is Z Z = ∑ { u , v } ⊆ V ( G ) [ d ( u ) · d ( v ) ] d ( u , v ) . If G is a tree, then both DD and ZZ are linearly related with the Wiener index W = ∑ { u , v } ⊆ V ( G ) d ( u , v ) . We examine the difference D D − Z Z for trees and establish a number of regularities. [ABSTRACT FROM AUTHOR]

We investigate the graph with minimal Zeroth-order general Randić index in terms of its order n , pendent number N 1 and cyclomatic number γ ≥ 0. Extremal graphs were completely characterized for cases of γ = 0 , 1 , 2 , which can be directly extended for graphs with cyclomatic number γ ≥ 3. [ABSTRACT FROM AUTHOR]