The differential on Graph Operator R(G)

TL;DR

This paper investigates the relationship between the differential of a graph G and its derived graph R(G), providing tight bounds and exploring connections with other graph parameters.

Contribution

It introduces bounds for the differential of R(G) based on the differential of G and explores related graph parameters, advancing understanding of graph transformations.

Findings

Derived bounds for $ ext{partial}(R(G))$ in terms of $ ext{partial}(G)$

Identifies relationships between vertex sets of G and R(G)

Connects differential with other graph parameters

Abstract

Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the number $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets $S\subseteq V(G)$ is the differential $\partial(G)$ of $G$. The graph $R{G}$ is defined as the graph obtained from $G$ by adding a new vertex $v_e$ for each $e\in E(G)$, and by joining $v_e$ to the end vertices of $e$. In this paper we study the relationship between $\partial(G)$ and $\partial(R(G))$, and give tight asymptotic bounds for $\partial(R(G))$. We also exhibit some relationships between certain vertex sets of $G$ and $R(G)$ which involve well known graph theoretical parameters.