Laplacian Energy of Generalized Complements of a Graph

H. J. Gowtham ; Sabitha D ; Souza ; Pradeep G. Bhat

Kragujevac Journal of Mathematics, Tome 42 (2018) no. 2, p. 299

Résumé

Let $P=\left\lbrace V_{1}, V_{2}, V_{3}, �, V_{k}\right\rbrace$ be a partition of vertex set $V(G)$ of order $k\geq 2$. For all $V_{i}$ and $V_{j}$ in $P$, $i\neq j$, remove the edges between $V_{i}$ and $V_{j}$ in graph $G$ and add the edges between $V_{i}$ and $V_{j}$ which are not in $G$. The graph $G_{k}^{P}$ thus obtained is called the extit{$k-$complement} of graph $G$ with respect to a partition $P$. For each set $V_{r}$ in $P$, remove the edges of graph $G$ inside $V_{r}$ and add the edges of $\overline{G}$ (the complement of $G$) joining the vertices of $V_{r}$. The graph $G_{k(i)}^{P}$ thus obtained is called the extit{$k(i)-$complement of graph } $G$ with respect to a partition $P$. In this paper, we study Laplacian energy of generalized complements of some families of graph. An effort is made to throw some light on showing variation in Laplacian energy due to changes in a partition of the graph.

Détail

Classification : 05C15 05C50
Keywords: Generalized complements, Laplacian spectrum, Laplacian energy