On the Number of Connected Edge Cover Sets of Some Graph Families

TL;DR

This paper derives explicit formulas for the connected edge cover polynomial and counts of such covers in various graph families, enhancing understanding of their combinatorial structures.

Contribution

It provides new explicit formulas and combinatorial proofs for the connected edge cover polynomial in several important graph families.

Findings

Explicit formulas for wheels, complete graphs, bipartite graphs, friendship, and lollipop graphs.

Verification of formulas through computational enumeration.

Enhanced understanding of the combinatorial structure of connected edge covers.

Abstract

Let $G=(V,E)$ be a simple connected graph. A connected edge cover of $G$ is a subset $S\subseteq E$ such that every vertex of $G$ is incident with at least one edge in $S$ and the subgraph induced by $S$ is connected. The connected edge cover polynomial of $G$ is defined as $E_c(G,x)=\sum_{i} e_c(G,i)x^i$, where $e_c(G,i)$ denotes the number of connected edge covers of $G$ with exactly $i$ edges. In this paper, we derive explicit formulas for both the connected edge cover polynomials and the total number of connected edge covers for several important graph families, including wheels, complete graphs $K_n$, complete bipartite graphs $K_{2,n}$, friendship graphs, and lollipop graphs. Each formula is accompanied by a combinatorial proof and verified by computational enumeration for small orders.