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

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.