Bolian Liu

On the nullity of bicyclic graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we obtain the nullity set of bicyclic graphs of order n, and determine the bicyclic graphs with maximum nullity.

On the nullity of graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is not isomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n− 2 and the second upper bound n− 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is also discussed. We obtain an upper bound of M(n, e), and characterize n and e for which the upper bound is achieved.

Matrices in Combinatorics and Graph Theory

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

The exponent set of symmetric primitive (0, 1) matrices with zero trace

Abstract We prove that the exponent set of symmetric primitive (0, 1) matrices with zero trace (the adjacency matrices of the simple graphs) is {2,3,…,2 n −4}⧹ S , where S is the set of all odd numbers in { n −2, n −1,…,2 n −5}. We also obtain a characterization of the symmetric primitive matrices with zero trace whose exponents attain the upper bound 2 n −4.

On the largest eigenvalue of non-regular graphs

We study the spectral radius of connected non-regular graphs. Let λ1(n,Δ) be the maximum spectral radius among all connected non-regular graphs with n vertices and maximum degree Δ. We prove that Δ−λ1(n,Δ)=Θ(Δ/n2). This improves two recent results by Stevanovic and Zhang, respectively.

The maximum Wiener polarity index of unicyclic graphs

Abstract The Wiener polarity index of a graph G is the number of unordered pairs of vertices u , v such that the distance between u and v is 3. In this paper, we obtain a upper bound for the Wiener polarity index of unicyclic chemical graphs. Moreover, the maximum Wiener polarity index of unicyclic graphs is determined, and the corresponding extremal graphs are characterized.

On the Wiener polarity index of trees with maximum degree or given number of leaves

The Wiener polarity index WP(G) of a graph G=(V,E) is the number of unordered pairs of vertices {u,v} of G such that the distance dG(u,v)=3. In this paper, the minimum (resp. maximum) Wiener polarity index of trees with n vertices and maximum degree @D are given, and the corresponding extremal trees are determined, where [emailxa0protected][emailxa0protected]@?n-1. Moreover, the trees minimizing WP(T) among all trees T of order n and k leaves are characterized, where [emailxa0protected][emailxa0protected]?n-1.

Some results on the Laplacian spectrum

A graph is called a Laplacian integral graph if the spectrum of its Laplacian matrix consists of integers, and a graph G is said to be determined by its Laplacian spectrum if there does not exist other non-isomorphic graph H such that H and G share the same Laplacian spectrum. In this paper, we obtain a sharp upper bound for the algebraic connectivity of a graph, and identify all the Laplacian integral unicyclic, bicyclic graphs. Moreover, we show that all the Laplacian integral unicyclic, bicyclic graphs are determined by their Laplacian spectra.

The polytope of even doubly stochastic matrices

We discuss some constraints for the polytope of even doubly stochastic matrices and investigate some of its other properties.

The second Zagreb indices of unicyclic graphs with given degree sequences

Let @p=(d1,d2,...,dn) and @p^=(d1^,d2^,...,dn^) be two different non-increasing degree sequences. We write @[emailxa0protected][emailxa0protected]^, if and only if @?i=1^nd[emailxa0protected]?i=1^ndi^, and @?i=1^jd[emailxa0protected][emailxa0protected]?i=1^jdi^ for all j=1,2,...,n. Let @C(@p) be the class of connected graphs with degree sequence @p. The second Zagreb index of a graph G is denoted by M2(G)[emailxa0protected]?uv@?E(G)d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if @[emailxa0protected][emailxa0protected]^, @p and @p^ are unicyclic degree sequences and U^* and U^*^* have the maximum second Zagreb indices in @C(@p) and @C(@p^), respectively, then M2(U^*)=17 vertices.