Lihua Feng

Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number☆

Abstract In this paper, we present sharp bounds for the Zagreb indices, Harary index and hyper-Wiener index of graphs with a given matching number, and we also completely determine the extremal graphs.

Minimizing the Laplacian spectral radius of trees with given matching number

Let denote the set of trees on n vertices with fixed matching number β. In this article, we prove that if n = kβ +1, k ≥ 2, then the trees which minimize the Laplacian spectral radius over have maximum degree Δ =k, and determine the extremal trees for 1≤ β ≤4.

Spectral radii of graphs with given chromatic number

Abstract We consider the set G n , k of graphs of order n with the chromatic number k ≥ 2 . In this note, we prove that in G n , k the Turan graph T n , k has the maximal spectral radius; and P n if k = 2 , C n if k = 3 and n is odd, C n − 1 1 if k = 3 and n is even, K k ( l ) if k ≥ 4 has the minimal spectral radius. Thus we answer a problem raised by Cao [D.S. Cao, Index function of graphs, J. East China Norm. Univ. Sci. Ed. 4 (1987) 1–8 (in Chinese). MR89m:05084] and Hong [Y. Hong, Bounds of eigenvalues of graphs, Discrete Math. 123 (1993) 65–74] in the affirmative.

The Signless Laplacian Spectral Radius of Unicyclic Graphs with Graph Constraints

In this paper, we study the signless Laplacian spectral radius of unicyclic graphs with prescribed number of pendant vertices or independence number. We also characterize the extremal graphs completely.

Signed graphs with small positive index of inertia

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

Further results on the distance spectral radius of graphs

Let D(G) be the distance matrix of a connected graph G. The distance spectral radius of G is the largest eigenvalue of D(G) and it has been proposed to be a molecular structure descriptor. In this article, we determine the unique trees with minimal and maximal distance spectral radii among trees with fixed bipartition. As a corollary, the trees with the first three minimal distance spectral radii are determined. Furthermore, we determine the unique trees with minimal distance spectral radii among n-vertex trees with fixed number of pendent vertices or fixed even diameter, respectively. We also propose a conjecture regarding the tree with minimal distance spectral radius among n-vertex trees with fixed odd diameter.

Wiener index, Harary index and graph properties

Abstract Finding sufficient conditions for some properties of graphs in light of quantitative methods is an important problem. In this paper, in terms of the Wiener index or Harary index, we present several sufficient conditions for a graph to be k -connected, β -deficient, k -hamiltonian, k -path-coverable or k -edge-hamiltonian.

Further results on the largest matching root of unicyclic graphs

Abstract Let G be a simple connected graph with vertex set V ( G ) . The matching polynomial of G is defined as M G ( x ) = ∑ k = 0 n ∕ 2 ( − 1 ) k m ( G , k ) x n − 2 k , where m ( G , k ) denotes the number of ways in which k independent edges can be selected in G . Let λ 1 ( G ) be the largest root of M G ( x ) . We determine the unicyclic graphs with the four largest and the two smallest λ 1 ( G ) -values.

The number of spanning trees of a graph with given matching number

The number of spanning trees of a graph G is the total number of distinct spanning subgraphs of G that are trees. In this paper, we present sharp upper bounds for the number of spanning trees of a graph with given matching number.

The signless Laplacian spectral radius of graphs on surfaces

Let G be an n-vertex simple graph embeddable on a surface of Euler genus γ. We present the upper bounds for the signless Laplacian spectral radius of G in terms of n and γ, with further improvements when G is an outerplanar or Halin graph.