On $\mathscr{P}$-energy of join of graphs

Abstract

Given a graph $G=(V, E)$ with a vertex partition $\mathscr{P}$ of cardinality $k$, we associate to it a real matrix $A_{\mathscr{P}}(G)$, whose diagonal entries are the cardinalities of elements in $\mathscr{P}$ and off-diagonal entries are from the set $\{2,1,0,-1\}$. The $\mathscr{P}$-energy $E_{\mathscr{P}}(G)$ is the sum of the absolute values of eigenvalues of $A \mathscr{P}(G)$. In this paper, we discuss $\mathscr{P}$-energy of the join of graphs using the concept of $M$-coronal of graphs and determine $\mathscr{P}$-energy for the complements of the join of graphs.