Monophonic Distance Laplacian Energy of Transformation Graphs Sn^++-,Sn^{+-+},Sn^{+++}

Abstract

Let $G$ be a simple connected graph of order $n$, $v_{i}$ its vertex. Let $\delta^{L}_{1}, \delta^{L}_{2}, \ldots, \delta^{L}_{n}$ be the eigenvalues of the distance Laplacian matrix $D^{L}$ of $G$. The distance Laplacian energy is denoted by $LE_{D}(G)$. This motivated us to defined the new graph energy monophonic distance Laplacian energy of graphs. The eigenvalues of monophonic distance Laplacian matrix $M^{L}\left(G\right)$ are denoted by $\mu^{L}_{1}, \mu^{L}_{2}, \ldots, \mu^{L}_{n}$ and are said to be $M^{L}$- eigenvalues of $G$ and to form the $M^{L}$-spectrum of $G$, denoted by $Spec_{M^{L}}(G)$. Here $MT_{G}\left(v_{j}\right)$ is the $j^{th}$ row sum of monophonic distance matrix of $M(G)$ and $\mu^{L}_{1}\leq\mu^{L}_{2}\leq, \ldots, \leq\mu^{L}_{n}$ be the eigenvalues of the monophonic distance Laplacian matrix is $M^{L}(G)$. The monophonic distance Laplacian energy is defined as $LE_{M}(G)$. In this paper we computed the monophonic distance Laplacian energy of $S^{++-}_{n}$, $S^{+-+}_{n}$, $S^{+++}_{n}$ graphs based on its spectrum values.\