On the j-Edge Intersection Graph of Cycle Graph

Abstract

This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called $j$-edge intersection graph of cycle graph, denoted by $E_{C_{(n,j)}}$. The vertex set of $E_{C_{(n,j)}}$ is the set of spanning subgraphs of cycle graph with $j$ edges where $n \geq 3$ and $j$ is a nonnegative integer such that $1 \leq j \leq n$. Moreover, two distinct vertices are adjacent if they have exactly one edge in common. $E_{C_{(n,j)}}$ is considered as a simple graph. Furthermore, $E_{C_{(n,j)}}$ is characterized by the value of $j$ that is when $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$ and $2 \leq j \leq \lceil \frac{n}{2} \rceil$. When $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$, the new graph only produced an empty graph. Hence, the proponents only considered the value when $2 \leq j \leq \lceil \frac{n}{2} \rceil$ in determining the order and size of $E_{C_{(n,j)}}$. Moreover, this paper discusses necessary and sufficient conditions where the $j$-edge intersection graph of $C_n$ is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of $E_{C_{(n,j)}}$ when $j=2$.