Unicyclic signed graphs with maximal energy

Abstract

Let $x_1,x_2,\dots ,x_n$ be the eigenvalues of a signed graph $\Gamma$ of order $n$. The energy of $\Gamma$ is defined as $E(\Gamma )={\sum }_{j=1}^n|x_j|.$ Let ${\mathcal{P}}_n^4$ be obtained by connecting a vertex of the negative circle $(C_4,\overline{\sigma })$ with a terminal vertex of the path $P_{n-4}$. In this paper, we show that for $n=4,6$ and $n\ge 8,$ ${\mathcal{P}}_n^4$ has the maximal energy among all connected unicyclic $n$-vertex signed graphs, except the cycles $C_5^{+},C_7^{+}.$