Ruifang Liu

Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U(n,d) be the set of unicyclic graphs on n vertices with diameter d. In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U(n,d). It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449-457].

The least eigenvalue of the signless Laplacian of non-bipartite unicyclic graphs with k pendant vertices

Let U(n,k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n � 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n,k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipartite unicyclic graphs on n vertices. As an application of the above results, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in Un is characterized, which has recently been proved by Cardoso, Cvetkovic, Rowlinson, and Simic. 1. Introduction. All graphs considered are simple, undirected, and connected. The vertex set and edge set of the graph G are denoted by V (G) and E(G), respec- tively. The distance between vertices u and v of a graph G is denoted by dG(u,v). The degree of a vertex v, written by dG(v) or d(v), is the number of edges incident with v. A pendant vertex is a vertex of degree 1. The set of the neighbors of a vertex v is denoted by NG(v) or N(v). The girth g(G) of a graph G is the length of the shortest cycle in G, with the girth of an acyclic graph being infinite. Denote by Cn

On the second largest distance eigenvalue of a graph

Let G be a simple connected graph of order n and D(G) be the distance matrix of G. Suppose that is the distance spectrum of G. A graph G is said to be determined by its D-spectrum if any graph with the same distance spectrum as G is isomorphic to G. In this paper, we consider spectral characterization on the second largest distance eigenvalue of graphs, and prove that the graphs with are determined by their D-spectra.

Graphs with small diameter determined by their D -spectra

Let G be a connected graph with vertex set V(G) = {v1, v2,..., vn}. The distance matrix D(G) = (dij)n×n is the matrix indexed by the vertices of G, where dij denotes the distance between the vertices vi and vj. Suppose that λ1(D) ≥ λ2(D) ≥... ≥ λn(D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n, d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n, d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].

Super λ 3 -optimality of regular graphs

Abstract Let G = ( V , E ) be a connected graph. An edge set S ⊂ E is a 3-restricted edge cut, if G − S is disconnected and every component of G − S has at least three vertices. The 3-restricted edge connectivity λ 3 ( G ) of G is the cardinality of a minimum 3-restricted edge cut of G . A graph G is λ 3 -connected, if 3-restricted edge cuts exist. A graph G is called λ 3 -optimal, if λ 3 ( G ) = ξ 3 ( G ) , where ξ 3 ( G ) = m i n { | [ X , X ¯ ] | : X ⊆ V , | X | = 3 , G [ X ] i s c o n n e c t e d } , [ X , X ¯ ] is the set of edges of G with one end in X and the other in X ¯ and X ¯ = V − X . Furthermore, if every minimum 3-restricted edge cut is a set of edges incident to a connected subgraph induced by three vertices, then G is said to be super 3-restricted edge connected or super- λ 3 for simplicity. In this paper we show that let G be a k -regular connected graph of order n ≥ 6 , if k ≥ ⌊ n / 2 ⌋ + 3 , then G is super- λ 3 .