Geodesic
Laplacian Energy of Union and Cartesian Product and Laplacian Equienergetic Graphs
Kragujevac Journal of Mathematics, Tome 39 (2015) no. 2, p. 193 .

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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.
Classification : 05C50, 05C75
Keywords: Laplacian spectrum, Laplacian energy, Cartesian product, Laplacian equienergetic graphs 
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     author = {Harishchandra S. Ramane and Gouramma A. Gudodagi and Ivan Gutman},
     title = {Laplacian {Energy} of {Union} and {Cartesian} {Product} and {Laplacian} {Equienergetic} {Graphs}},
     journal = {Kragujevac Journal of Mathematics},
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     url = {http://geodesic.mathdoc.fr/item/KJM_2015_39_2_a7/}
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Harishchandra S. Ramane; Gouramma A. Gudodagi; Ivan Gutman. Laplacian Energy of Union and Cartesian Product and Laplacian Equienergetic Graphs. Kragujevac Journal of Mathematics, Tome 39 (2015) no. 2, p. 193 . http://geodesic.mathdoc.fr/item/KJM_2015_39_2_a7/