Rundan Xing

On the revised Szeged index

We give bounds for the revised Szeged index, and determine the n-vertex unicyclic graphs with the smallest, the second-smallest and the third-smallest revised Szeged indices for n>=5, and the n-vertex unicyclic graphs with the kth-largest revised Szeged indices for all k up to 3 for n=5, to 5 for n=6, to 6 for n=7, to 7 for n=8, and to @?n2@?+4 for n>=9. We also determine the n-vertex unicyclic graphs of cycle length r, 3@?r@?n, with the smallest and the largest revised Szeged indices.

On Atom-Bond Connectivity Index

The atom-bond connectivity (ABC) index, introduced by Estrada et al. in 1998, displays an excellent correlation with the formation heat of alkanes. We give upper bounds for this graph invariant using the number of vertices, the number of edges, the Randi´c connectivity indices, and the first Zagreb index. We determine the unique tree with the maximum ABC index among trees with given numbers of vertices and pendant vertices, and the n-vertex trees with the maximum, and the second, the third, and the fourth maximum ABC indices for n ≥ 6.

On the distance signless Laplacian spectral radius of graphs

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 where is the diagonal matrix of vertex transmissions of G and is the distance matrix of G. In this paper, we determine the graphs with minimum distance signless Laplacian spectral radius among the trees, unicyclic graphs and bipartite graphs with fixed numbers of vertices, respectively, and determine the graphs with minimum distance signless Laplacian spectral radius among the connected graphs with fixed numbers of vertices and pendant vertices, and the connected graphs with fixed number of vertices and connectivity, respectively.

On the second largest distance eigenvalue

We characterize all connected graphs whose second largest distance eigenvalues belong to , as well as all trees whose second distance eigenvalues belong to . We also consider unicyclic graphs whose second distance eigenvalues belong to .We consider simple undirected graphs. Let G be a connected graph with vertex set V (G) = {v1, . . . , vn}. For 1 ≤ i, j ≤ n, the distance between vertices vi and vj in G, denoted by dG(vi, vj) or simply dvivj , is the length of a shortest path connecting them in G. The distance matrix of G is the n × n matrix D(G) = (dvivj ). Since D(G) is symmetric, the eigenvalues of D(G) are all real numbers. The distance eigenvalues of G, denoted by λ1(G), . . . , λn(G), are the eigenvalues of D(G), arranged in nonincreasing order. For 1 ≤ k ≤ n, we call λk(G) the kth distance eigenvalue of G. The study of distance eigenvalues dates back to the classical work of Graham and Pollack [4], Edelberg et al. [2] and Graham and Lovász [3] in 1970s. Merris [8] studied the relations between the distance eigenvalues and the Laplacian eigenvalues of trees. The first distance eigenvalue has received much attention. Ruzieh and Powers [10] showed that the path Pn is the unique n-vertex connected graph with maximal first distance eigenvalue, while the complete graph Kn is the unique n-vertex connected graph with minimal first distance eigenvalue. Among others, Stevanović and Ilić [11] showed that the star Sn is the unique n-vertex tree with minimal first distance eigenvalue. The extremal graphs with maximal or minimal first distance eigenvalues may be found in, e.g., [1, 9, 12, 13, 15]. The last (least) distance eigenvalue has also received some attention, see [6, 14]. Let G1 and G2 be two vertex-disjoint graphs. G1 ∪G2 denotes the vertex-disjoint union of G1 and G2, and G1∨G2 denotes the graph obtained from G1∪G2 by joining each vertex of G1 and each vertex of G2 using an edge.

On the two largest distance eigenvalues of graph powers

We give sharp lower bounds for the largest and the second largest distance eigenvalues of the k-th power of a connected graph, determine all trees and unicyclic graphs for which the second largest distance eigenvalues of the squares are less than 5 - 3 2 , and determine the unique n-vertex trees of which the squares achieve minimum and second-minimum largest distance eigenvalues, as well as the unique n-vertex trees of which the squares achieve minimum, second-minimum and third-minimum second largest distance eigenvalues. Lower bounds for first two largest distance eigenvalues of graph powers are given.All trees and unicyclic graphs G with λ 2 ( G 2 ) < 5 - 3 2 are determined.Unique n-vertex trees G with small λ 1 ( G 2 ) and λ 2 ( G 2 ) are determined.

A note on distance spectral radius of trees

Abstract The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

On hyper-Kirchhoff index

Abstract The hyper-Kirchhoff index is introduced when the hyper-Wiener operator is applied to the resistance-distance matrix of a connected graph. We give lower and upper bounds for the hyper-Kirchhoff index, and determine the n -vertex unicyclic graphs with the smallest, the second and the third smallest as well as the largest, the second and the third largest hyper-Kirchhoff indices for n ≥ 5 . We also determine the n -vertex unicyclic graphs of cycle length s , 3 ≤ s ≤ n , with the smallest and the largest hyper-Kirchhoff indices.