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

Diana R, Binu Selin T


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.\


distance Laplacian energy,Monophonic distance matrix,Monophonic distance Laplacian eneergy

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Wu, Bayoyindureng. "Basic Properties of Total Transformation Graphs." Journal of Mathematical Study, Vol. 24(2), 109-116, (2001).

R. Diana, T. Binu Selin. "Monophonic Distance Laplacian Energy of Graphs." Advances and Applications in Mathematical Sciences, Vol. 21(7), 3865-3872, (2022).

Gopalapillai Indulal, I. Gutman, Ambat Vijayakumar. "On Distance Energy of Graphs." MATCH Commun. Math. Comput., Vol. 60(2), 461-472, (2008).

I. Gutman. "The Energy of a Graph." Besmath-statist.sekt.Forschungsz.graz, Vol. 103, 1-22, (1978).

F. Harary. "Graph Theory." Addison-Wesley, Boston, (1969).

S. Harish Chandra, Ivan Gutman. "Seidal Signless Laplacian Energy of Graphs." Mathematics Interdisciplinary Research, Vol. 2, 181-191, (2017).

Ivan Gutman, Bo Zhou. "Laplacian Energy of a Graph." Linear Algebra and its Applications, Vol. 414, 29-37, (2006).

Jieshan Yang, Lihua You, I. Gutman. "Bounds on the Distance Laplacian Energy of Graphs." Kragujevac Journal of Mathematics, 37(2), 245-255, (2013).

H. S. Ramane, R. B. Jummannaver, Ivan Gutman. "Seidal Laplacian Energy of Graphs." International Journal of Applied Graph Theory, Vol. 1(2), 74-82, (2017).

A. P. Santhakumaran, P. Titus, K. Ganesamoorthy. "Monophonic Distance in Graphs." Discrete Mathematics, Algorithms and Applications, Vol. 3(2), 159-169, (2011).

V. S. Shigehalli, Kenchappa S Betageri. "Color Laplacian Energy of Graphs." Journal of Computer and Mathematical Sciences, Vol. 6(9), 485-494, (2015).

Shridhar Chandrakant Patekar, Maruti Mukinda Shikare. "On Path Laplacian Eigenvalues and Path Laplacian Energy of Graphs." Journal of New Theory, Vol. 20, 93-101, (2018).



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