Some Results On Spectrum And Energy Of Graphs With Loops

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