Laplacian Energy of Union and Cartesian Product and Laplacian Equienergetic Graphs

Harishchandra S. Ramane, Gouramma A. Gudodagi, Ivan Gutman

The Laplacian energy of a graph $G$ with $n$ vertices and $m$ edges is defined as $LE(G) = \sum_{i=1}^{n}\left|\mu_i - 2m/n \right|$, where $\mu_1,\mu_2,\ldots,\mu_n$ are the Laplacian eigenvalues of $G$. If two graphs $G_1$ and $G_2$ have equal average vertex degrees, then $LE(G_1 \cup G_2) = LE(G_1) + LE(G_2)$. Otherwise, this identity is violated. We determine a term $\Xi$, such that $LE(G_1) + LE(G_2)-\Xi \leq LE(G_1 \cup G_2) \leq LE(G_1) + LE(G_2) +\Xi$ holds for all graphs. Further, by calculating $LE$ of the Cartesian product of some graphs, we construct new classes of Laplacian non-cospectral, Laplacian equienergetic graphs.