# The differential on operator S(Γ)

Math Biosci Eng. 2023 May 5;20(7):11568-11584. doi: 10.3934/mbe.2023513.

## Abstract

Consider a simple graph $\Gamma = (V(\Gamma), E(\Gamma))$ with $n$ vertices and $m$ edges. Let $P$ be a subset of $V(\Gamma)$ and $B(P)$ the set of neighbors of $P$ in $V(\Gamma)\backslash P$. In the study of graphs, the concept of differential refers to a measure of how much the number of edges leaving a set of vertices exceeds the size of that set. Specifically, given a subset $P$ of vertices, the differential of $P$, denoted by $\partial(P)$, is defined as $|B(P)|-|P|$. The differential of $\Gamma$, denoted by $\partial(\Gamma)$, is then defined as the maximum differential over all possible subsets of $V(\Gamma)$. Additionally, the subdivision operator ${{\mathcal{S}}({\Gamma})}$ is defined as the graph obtained from $\Gamma$ by inserting a new vertex on each edge of $\Gamma$. In this paper, we present results for the differential of graphs on the subdivision operator ${{\mathcal{S}}({\Gamma})}$ where some of these show exact values of $\partial({{\mathcal{S}}({\Gamma})})$ if $\Gamma$ belongs to a classical family of graphs. We obtain bounds for $\partial({{\mathcal{S}}({\Gamma})})$ involving invariants of a graph such as order $n$, size $m$ and maximum degree $\Delta$, and we study the realizability of the graph $\Gamma$ for any value of $\partial({{\mathcal{S}}({\Gamma})})$ in the interval $\left[n-2, \frac{n(n-1)}{2}-n+2\right]$. Moreover, we give a characterization for $\partial({{\mathcal{S}}({\Gamma})})$ using the notion of edge star packing.

Keywords: differential of graphs; independence number; matching number; subdivision graph.