Complementary Vanishing Graphs

TL;DR

This paper investigates the class of graphs called complementary vanishing graphs, characterizing them through combinatorial conditions and minimal examples, and determines all such graphs with up to 8 vertices.

Contribution

It introduces the concept of complementary vanishing graphs, provides combinatorial characterizations, and classifies all such graphs on at most 8 vertices.

Findings

Characterization of when a graph is complementary vanishing.

Identification of minimal complementary vanishing graphs.

Complete classification of graphs with up to 8 vertices that are complementary vanishing.

Abstract

Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\ne j$ we have $a_{i,j}\ne 0$ if and only if $v_iv_j\in E(G)$. Motivated by the Graph Complement Conjecture, we say that a graph $G$ is complementary vanishing if there exist matrices $A \in \mathcal{S}(G)$ and $B \in \mathcal{S}(\overline{G})$ such that $AB=O$. We provide combinatorial conditions for when a graph is or is not complementary vanishing, and we characterize which graphs are complementary vanishing in terms of certain minimal complementary vanishing graphs. In addition to this, we determine which graphs on at most $8$ vertices are complementary vanishing.