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arXiv · 2304.05275

Some Results On Spectrum And Energy Of Graphs With Loops

Saieed Akbari·Hussah Al Menderj·Miin Huey Ang·Johnny Lim·Zhen Chuan Ng
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Abstract

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($G_{V(G)\backslash S}$). We also prove that for every $S \subseteq V(G)$, $E(G_S) \geq E(G)$ 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 $G_S$.

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Saieed Akbari, Hussah Al Menderj, Miin Huey Ang, Johnny Lim, Zhen Chuan Ng. 2023-04-11. Some Results On Spectrum And Energy Of Graphs With Loops. https://doi.org/10.1007/s40840-023-01489-z

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