On the Extremal Energy of Complex Unit Gain Dumbbell Graphs

Abstract

We study the extremal energy problem for complex unit gain graphs whose underlying graph is the dumbbell graph $D_{r,s,\ell}$. Using switching equivalence, we reduce the spectrum to the real parts of the two cycle gains and obtain an explicit expression of the characteristic polynomial in terms of matching polynomials of natural subgraphs. For the bipartite case, we determine the extremal gain assignments by coefficient comparison. For the non-bipartite cases, we analyze the Coulson integral kernels. Finally, the maximum-energy conditions are determined in all cases, while the minimum-energy conditions are determined except when $r$, $s$, and $\ell$ are all odd. For this remaining case, we alternatively prove sign restrictions for any improvement over $(0,0)$, and prove a Hessian criterion at the origin, which provides a sufficient condition for $(0,0)$ to fail to be an energy minimizer.