Parity of an odd dominating set

TL;DR

This paper establishes a relationship between the parity of odd dominating sets in a graph and the rank of its closed neighborhood matrix, leading to new insights and formulas for graph nullity.

Contribution

It introduces a novel connection between the parity of odd dominating sets and the rank of the closed neighborhood matrix, providing new theoretical results.

Findings

Parity of odd dominating sets equals the rank of the graph

Derived a formula for nullity of graph joins

Established properties of the closed neighborhood matrix

Abstract

For a simple graph $G$ with vertex set $V(G)=\{v_1,...,v_n\}$, we define the closed neighborhood set of a vertex $u$ as $N[u]=\{v \in V(G) \; | \; v \; \text{is adjacent to} \; u \; \text{or} \; v=u \}$ and the closed neighborhood matrix $N(G)$ as the matrix obtained by setting to $1$ all the diagonal entries of the adjacency matrix of $G$. We say a set $S$ is odd dominating if $N[u]\cap S$ is odd for all $u\in V(G)$. We prove that the parity of an odd dominating set of $G$ is equal to the parity of the rank of $G$, where the rank of $G$ is defined as the dimension of the column space of $N(G)$. Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.