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Abstract: In 1970s, Gutman introduced the concept of the energy $\EnG$ for a simplegraph $G$, which is defined as the sum of the absolute values of theeigenvalues of $G$. This graph invariant has attracted much attention, and manylower and upper bounds have been established for some classes of graphs amongwhich bipartite graphs are of particular interest. But there are only a fewgraphs attaining the equalities of those bounds. We however obtain an exactestimate of the energy for almost all graphs by Wigner-s semi-circle law, whichgeneralizes a result of Nikiforov. We further investigate the energy of randommultipartite graphs by considering a generalization of Wigner matrix, andobtain some estimates of the energy for random multipartite graphs.



Author: Wenxue Du, Xueliang Li, Yiyang Li

Source: https://arxiv.org/



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