Ordering of the unicyclic signed graphs with perfect matchings by their minimal energies

Abstract

Let $\mathcal {U}_{2n}^{\sigma}$ be the set of the unicyclic signed graphs with perfect matchings having $2n$ vertices, where $\sigma$ is a signing function from the edge set of the graphs considered to $\{-1,1\}$.   The increasing order of the signed graphs among $\mathcal {U}_{2n}^{\sigma}$ according to their minimal energies is considered.  A relationship between the energies of a unicyclic graph and of its signed graphs is derived.  A new integral formula for comparing the energies of two signed graph is introduced. In $\mathcal {U}_{2n}^{\sigma}$ with $n\geq 721$, the first  18 signed  graphs in the increasing order by their minimal energies  are obtained.