On the number of connected edge cover sets in a graph

TL;DR

This paper introduces the concept of the connected edge cover polynomial, which counts connected edge cover sets in a graph, and explores its properties and specific cases.

Contribution

It initiates the study of the connected edge cover polynomial and analyzes its properties for certain classes of graphs.

Findings

Defined the connected edge cover polynomial $E_c(G, i)$.

Derived initial properties and formulas for the polynomial.

Analyzed the polynomial for specific graph classes.

Abstract

Let $ G=(V,E) $ be a simple graph of order $ n $ and size $ m $. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is connected. We initiate the study of the number of the connected edge cover sets of a graph $G$ with cardinality $i$, $ e_{c}(G,i) $ and consider the generating function for $ e_{c}(G,i) $ which is called the connected edge cover polynomial of $ G $. After obtaining some results for this polynomial, we investigate this polynomial for some certain graphs.