An upper bound for the nullity of a bipartite graph in terms of its maximum degree

Abstract

Let G be a connected undirected graph without loops and multiple edges. The graph G is said to be reduced if distinct vertices of G have distinct neighbours in G. The nullity η(G) of G is the multiplicity of 0 as an eigenvalue of the adjacency matrix of G. By |G| and (G), we respectively denote the order and the maximum degree of G. In this note, it is proved that η(G) ≤ |G| − 2 − 2 ln 2 ( (G)) if G is a reduced bipartite graph and the graphs attaining equality are characterized.

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

“…This document presents a selection of current and significant papers, highlighting their respective value. The concept of nullity is quantified by the greatest degree of a vertex 8 , 9 . The literature contains scholarly works that go into the realm of pure mathematics and abstract theory on nullity, which may be found in the references 10 – 12 .…”

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

“…Pure mathematical and abstract theory on nullity is available in [19] , [20] , [21] . Nullity is expressed in terms of maximum degree of a vertex [22] , [23] . Authors of [24] , [25] , computed the double metric resolvability of convex polytopes, authors of [26] , computed the edge version of resolvability and double resolvability of some generalized graphs.…”

Proof of a conjecture on the nullity of a graph

“…In 1957, Collatz and Sinogowitz [7] posed the problem of characterizing all singular graphs. The problem has received considerable attention in the literature (see, eg, [3‐5,10‐12,14‐21,24,26‐31,33] and references therein). In studying the above problem, some attentions are attracted to bound the nullity of a graph by using some of the structure parameters, such as the order, the matching number, the number of pendant vertices, and especially the maximum degree of the graph (see [5,6,8,12,14,15,17,22‐24,29,30] for details).…”

“…A connect graph in which distinct vertices have distinct neighborhoods is called reduced. For a reduced bipartite graph, Song et al [28] improved the above bound to .…”

“…Proposition 1.3 (Song et al [28,Theorem 2.2]). Let G be a reduced bipartite graph with n vertices and with Δ as maximum degree.…”