Some Results on Spectrum and Energy of Graphs with Loops

Let $$G_S$$ be a graph with loops obtained from G of order n and at $$S \subseteq V(G).$$ In this paper, we establish neccesary sufficient condition on the bipartititeness connected spectrum $${\textrm{Spec}}(G_S)$$ $${\textrm{Spec}}(G_{V(G)\backslash S})$$ . We also prove that for every $$S\subseteq V(G),$$ $${\mathcal E}(G_S) \ge {\mathcal E}(G)$$ when is bipartite. Moreover, provide an identification complete graphs $$K_n$$ bipartite $$K_{m,n}$$ loops. characterize any whose eigenvalues are all positive or non-negative, few distinct eigenvalues. Finally, some bounds related to