The differential on Graph Operator $\S{G}$

TL;DR

This paper investigates the relationship between the differential of a graph and the differential of its associated graph $ ilde{G}$, also relating it to domination and independence parameters.

Contribution

It introduces the graph $ ilde{G}$ based on the incidence between vertices and edges and explores how the differential of $G$ relates to that of $ ilde{G}$, connecting it with known graph parameters.

Findings

Established bounds between $ ext{partial}(G)$ and $ ext{partial}( ilde{G})$

Linked the differential to domination and independence numbers

Provided new insights into graph parameter relationships

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 defined as $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets $S\subseteq V$ is the differential $\partial(G)$ of $G$. A graph operator is a mapping $F: G\rightarrow G'$, where $G$ and $G'$ are families of graphs.The graph $\S{G}$ is defined as the graph obtained from $G$ con bipartici\'on de v\'ertices $V(G)\cup E(G)$, donde hay tantas aristas entre $v \in V(G)$ y $e \in E(G)$, como veces $e$ sea incidente con $v$ en $G$. In this paper we study the relationship between $\partial(G)$ and $\partial(\S{G})$. Besides, we relate the differential of a graph with known parameters of a graph, namely, its domination and independence number.