The maximum index and spectral radius of unbalanced signed multipartite graphs

Let $\Gamma=(G,\sigma)$ be a signed graph, where $G$ is the underlying graph with vertex set $V(G)$ and edge set $E(G)$ such that $\sigma: E(G)\to \{-1,1\}$ is the sign function. For $U\subset V(G)$, the operation that changes the sign of all edges between $U$ and $V(G)\setminus U$ is called switching. Two signed graphs with the same underlying graph are switching equivalent if one is obtainable from the other one by switching a subset. Two signed graphs are switching isomorphic if one is isomorphic to a switching equivalent signed graph of the other one. A signed cycle is called negative if it contains an odd number of negative edges. A signed graph is balanced if none of its cycles is negative; otherwise it is unbalanced. The adjacency matrix $A(\Gamma)$ of $\Gamma$ is obtained from the standard $(0,1)$-adjacency matrix of $G$ by reversing the sign of all $1$s which correspond to negative edges. The index of $\Gamma$ is the largest eigenvalue of $A(\Gamma)$ and the spectral radius of $\Gamma$ is the largest absolute value of the eigenvalue of $A(\Gamma)$. The least eigenvalue of $\Gamma$ is the least eigenvalue of $A(\Gamma)$. We study the extremal problems of the index and the spectral radius among unbalanced signed multipartite graphs. More precisely, we determine the unbalanced signed $t$-partite graphs with fixed $t\ge 2$ and partite sizes (order, respectively) that maximizes the index and the spectral radius respectively, up to switching isomorphism. To determine the unbalanced signed multipartite graphs with fixed partite sizes (order, respectively) with maximum spectral radius, we also determine those with minimum least eigenvalue.