General Quasi-Laplacian Matrix of Weighted Mixed Pseudograph

Abstract

Many practical problems can be described as a kind graph model having multiple weighted directed edges, undirected edges and weighted vertices simultaneously however, which are lack of sufficient considerations in previous literatures. We discuss some basic definitions for this more general graph and derive some relevant results for its quasi-Laplacian spectrum. Basic non negative results of its eigenvalues are proposed. Furthermore, we argue some dynamic properties between graph spectral layout and its structural variation. Here two types networks evolutionary manner involving insert of new edge or new vertex are considered. The strictness of related conclusions is guaranteed by theoretical analysis. These initial investigation of weighted mixed pseudograph maybe conduct potential theoretical or applied values for a numerous of practical problems.