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Ordering of minimal energies in unicyclic signed graphs
Tahir Shamsher, Mushtaq A. Bhat, Shariefuddin Pirzada, and Yilun Shang
Volume 65, no. 1 (2023), pp. 119–133    

https://doi.org/10.33044/revuma.2565

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Abstract

Let $S=(G,\sigma)$ be a signed graph of order $n$ and size $m$ and let $t_1,t_2,\dots,t_n$ be the eigenvalues of $S$. The energy of $S$ is defined as $E(S)=\sum_{j=1}^{n}|t_j|$. A connected signed graph is said to be unicyclic if its order and size are the same. In this paper we characterize, up to switching, the unicyclic signed graphs with first $11$ minimal energies for all $n \geq 11$. For $3\leq n \leq 7$, we provide complete orderings of unicyclic signed graphs with respect to energy. For $8 \leq n \leq 10$, we determine unicyclic signed graphs with first $13$ minimal energies.

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