布尔矩阵平方根问题及其与图着色问题的关系

讨论了布尔矩阵平方根问题及其与图着色问题的关系.首先得到有平方根的布尔矩阵具有的一些性质;然后给出布尔矩阵存在平方根的一个充要条件;最后证明布尔矩阵的平方根问题可以转化为简单图的着色问题.     

图P^3n的奇优美标号算法

本文讨论了图P^3n的奇优美性,给出了图只奇优美标号算法．     

L（1, 2）-edge-labelings for lattices

For a graph G and two positive integers j and k, an m-L(j, k)-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by kThe λ j,k-number of G is the minimum m such that an m-L(j, k)-edge-labeling is admitted by GIn this article, the L(1, 2)-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λ j,k-numbers of these graphs are obtained.     

A note on unimodular congruence of the Laplacian matrix of a graph

Let G be a simple connected graph and L(G) be its Laplacian matrix. In this note, we prove that L(G) is congruent by a unimodular matrix to its Smith normal form if and only if G is a tree.     

Percolation in the secrecy graph

The secrecy graph is a random geometric graph which is intended to model the connectivity of wireless networks under secrecy constraints. Directed edges in the graph are present whenever a node can talk to another node securely in the presence of eavesdroppers, which, in the model, is determined solely by the locations of the nodes and eavesdroppers. In the case of infinite networks, a critical parameter is the maximum density of eavesdroppers that can be accommodated while still guaranteeing an infinite component in the network, i.e., the percolation threshold. We focus on the case where the locations of the nodes and eavesdroppers are given by Poisson point processes, and present bounds for different types of percolation, including in-, out- and undirected percolation.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.