Kexiang Xu

The Zagreb indices of graphs with a given clique number

Abstract For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Let W n , k be the set of connected n -vertex graphs with clique number k . In this work we characterize the graphs from W n , k with extremal (maximal and minimal) Zagreb indices, and determine the values of corresponding indices.

Trees with the seven smallest and eight greatest Harary indices

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, we determined the first up to seventh smallest Harary indices of trees of order n>=16 and the first up to eighth greatest Harary indices of trees of order n>=14.

On the Harary index of graph operations

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, expressions for the Harary indices of the join, corona product, Cartesian product, composition and disjunction of graphs are derived and the indices for some well-known graphs are evaluated. In derivations some terms appear which are similar to the Harary index and we name them the second and third Harary index.MSC:05C05, 05C07, 05C90.

On the Hosoya index and the Merrifield–Simmons index of graphs with a given clique number

Abstract The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let W n , k be the set of connected graphs with n vertices and clique number k . In this note we characterize the graphs from W n , k with extremal (maximal and minimal) Hosoya indices and the ones with extremal (maximal and minimal) Merrifield–Simmons indices, respectively.

Extremal t-apex trees with respect to matching energy

The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. For any integer t≥1, a graph G is called t-apex tree if there exists a t-set X⊆V(G) such that G − X is a tree, while for any Y⊆V(G) with |Y|<t, G − Y is not a tree. Let Tt(n) be the set of t-apex trees of order n. In this article, we determine the extremal graphs from Tt(n) with minimal and maximal matching energies, respectively. Moreover, as an application, the extremal cacti of order n and with s cycles have been completely characterized at which the minimal matching energy are attained.

The Harary Index of a graph

Introduction.- Extremal Graphs with Respect to Harary Index.- Relation Between the Harary Index and Related Topological Indices.- Some Properties and Applications of Harary Index.- The Variants of Harary Index.- Open Problems.

Weighted Harary indices of apex trees and k -apex trees

If G is a connected graph, then H A ( G ) = ? u ? v ( deg ( u ) + deg ( v ) ) / d ( u , v ) is the additively Harary index and H M ( G ) = ? u ? v deg ( u ) deg ( v ) / d ( u , v ) the multiplicatively Harary index of G . G is an apex tree if it contains a vertex x such that G - x is a tree and is a k -apex tree if k is the smallest integer for which there exists a k -set X ? V ( G ) such that G - X is a tree. Upper and lower bounds on H A and H M are determined for apex trees and k -apex trees. The corresponding extremal graphs are also characterized in all the cases except for the minimum k -apex trees, k ? 3 . In particular, if k ? 2 and n ? 6 , then H A ( G ) ? ( k + 1 ) ( 3 n 2 - 5 n - k 2 - k + 2 ) / 2 holds for any k -apex tree G , equality holding if and only if G is the join of K k and K 1 , n - k - 1 .

Extremal Laplacian-energy-like invariant of graphs with given matching number

Abstract. Let G be a graph of order n with Laplacian spectrum µ 1 ≥ µ 2 ≥ ··· ≥ µ n . TheLaplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n P −1k=1 √µ k . In thisnote, the extremal (maximal and minimal) LEL among all the connected graphs with given matchingnumber is determined. The corresponding extremal graphs are completely characterized with respectto LEL. Moreover a relationship between LEL and the independence number is presented in this note.Key words. Laplacian matrix, Laplacian-energy-like, Matching number.AMS subject classifications. 05C50, 15A18. 1. Introduction. Let G = (V,E) be a simple undirected graph with vertexset V (G) = {v 1 ,v 2 ,v 3 ,...,v n } and edge set E(G). Also let d i be the degree of thevertex v i for i = 1,2,...,n. Assume that A(G) is the (0,1)-adjacency matrix of Gand D(G) is the diagonal matrix of vertex degrees. The Laplacian matrix of G isL(G) = D(G) − A(G). The Laplacian polynomial P(G,λ) of G is the characteristicpolynomial of its Laplacian matrix, P(G,λ) = det(λI

Some extremal graphs with respect to inverse degree

The inverse degree of graph G is defined as I D ( G ) = ? v ? V ( G ) 1 d G ( v ) where d G ( v ) is the degree of vertex v in G . In this paper we have determined some upper and lower bounds on the inverse degree I D ( G ) for a connected graph G in terms of other graph parameters, such as chromatic number, clique number, connectivity, number of cut edges, matching number. Also the corresponding extremal graphs have been completely characterized.

Ordering connected graphs by their Kirchhoff indices

The Kirchhoff index of a graph G is 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 deleting p edges from a complete graph with and 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 .