Proof of a conjecture on the nullity of a graph

Abstract

Let G be a finite undirected graph without loops and multiple edges. The nullity of G, written as η G (), is defined to be the multiplicity of 0 as an eigenvalue of its adjacency matrix. The left problem of establishing How to cite this article: Wang L, Geng X. Proof of a conjecture on the nullity of a graph.

Study of some graph theoretical parameters for the structures of anticancer drugs

“…The paper by 22 presents a study on the correlation between the matching number and rank of a graph. The hypothesis about nullity is supported by a proof presented in the work of 23 .…”

Some stable and closed-shell structures of anticancer drugs by graph theoretical parameters

“…Only recent and few important articles are given herewith their importance. A proof is given on the conjecture on the topic of nullity [8] . Rank four graphs are characterized in [9] , rank five in [10] .…”

Geometric vs Algebraic Nullity for Hyperpaths

“…One may combine these two questions by asking, "What is the combinatorial meaning of each notion of the multiplicity of the zero eigenvalue of (hyper)graphs?" For the Laplacian matrix L(G) = D(G) − A(G) of a graph, in the 1970s, Fiedler showed that the multiplicityin both the algebraic and geometric senses -of the zero eigenvalue is equal to the number of components of G. Thus it is natural to ask this same question about the seemingly simpler adjacency matrix A(G), and indeed considerable attention has been given to Question 1 (e.g., [4,6,7,11,13]). Because A(G) is real symmetric and therefore diagonalizable, the answer to Question 2 is simple for a graph, however: they agree.…”