On the double covers of a line graph

TL;DR

This paper explores the properties of double covers of line graphs, introduces a new symmetric edge graph, and demonstrates the existence of non-cospectral equienergetic graphs using these concepts.

Contribution

It defines the symmetric edge graph as a new double cover of line graphs and investigates its properties and relationships with other double covers.

Findings

$L(X^{ ext{''}})$ is a double cover of $L(X)$

$ ext{γ}(X)$ is a new double cover of $L(X)$

Existence of non-cospectral equienergetic graphs for $k \\geq 5$

Abstract

Let $L(X)$ be the line graph of graph $X$. Let $X^{\prime\prime}$ be the Kronecker product of $X$ by $K_2$. In this paper, we see that $L(X^{\prime\prime})$ is a double cover of $L(X)$. We define the symmetric edge graph of $X$, denoted as $\gamma(X)$ which is also a double cover of $L(X)$. We study various properties of $\gamma(X)$ in relation to $X$ and the relationship amongst the three double covers of $L(X)$ that are $L(X^{\prime\prime}),\gamma(X)$ and $L(X)^{\prime\prime}$. With the help of these double covers, we show that for any integer $k\geq 5$, there exist two equienergetic graphs of order $2k$ that are not cospectral.