Maximizing the indices of a class of signed complete graphs

The index of a signed graph is the largest eigenvalue of its adjacency matrix. Let $\mathfrak{U}_{n,k,4}$ be the set of all signed complete graphs of order $n$ whose negative edges induce a unicyclic graph of order k and girth at least $4$. In this paper, we identify the signed graphs achieving the maximum index in the class $\mathfrak{U}_{n,k,4}$.