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arXiv · 2208.13462

Minimizers for the energy of eccentricity matrices of trees

Iswar Mahato·M. Rajesh Kannan
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Abstract

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star. Also, we characterize the trees with minimum $\mathcal{E}$-energy among all trees on $n$ vertices.

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Iswar Mahato, M. Rajesh Kannan. 2022-08-29. Minimizers for the energy of eccentricity matrices of trees. https://arxiv.org/abs/2208.13462

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