Some Research On The Rank Of Simple Graphs And Signed Graphs

The theory of graph spectra is an active and important field in graph theory, theresearch on the rank (or nullity) of the graph has become a hot topic in spectral graphtheory. In1957, L. Collatz and U. Sinogowitz posed the problem of characterizing allsingular graphs or nonsingular graphs. This problem has received much attention in m-athematics.A signed graphΓ=Γ(G,σ), consists of a underlying graphσ: E(G)-,(+,-). and an edgesign function, which assigns each edge of G a sign, either positive or negative. In1953, signed graphs were introduced by Harary in connection with the study of the the-ory of social balance in social psychology. Subsequently, a lot of problems of graphswere extended to signed graphs, such as the rank (or nullity) of signed graphs, coloringof signed graphs, planar signed graphs and so on.In this thesis, we mainly discuss some problems on the rank of simple graphs andsigned graphs. The main contents of the thesis are as follows: We mainly introduce thedevelopment of graph theory, the research background of this thesis and some basicterminologies and notations of graph theory in chapter one. We characterize all the lon-g graphs with rankn, for arbitrary positive integernin chapter two. We characterize allthe connected and reduced signed graphs with rank2or3and the connected and reduc-ed planar signed graphs with rank4in chapter three. We characterize all the reducedplanar signed long graphs with rank n, for arbitrary positive integer n in chapter four.We primarily summarize the main conclusions of the thesis and give some considerableproblems for further study in chapter five.