Maximal signed bipartite graphs with totally disconnected graphs as star complements

Let $\dot{\mathscr{B}}\triangleq \dot{\mathscr{B}}_{\dot{H}, \mu}$ denote an arbitrary signed bipartite graph with $\dot{H}$ as a star complement for an eigenvalue $\mu$, where $\dot{H}$ is a totally disconnected graph of order $s$. In this paper, by using Hadamard and Conference matrices as tools, the maximum order of $\dot{\mathscr{B}}$ and the extremal graphs are studied. It is shown that $\dot{\mathscr{B}}$ exists if and only if $\mu^2$ is a positive integer. A formula of the maximum order of $\dot{\mathscr{B}}$ is given in the case of $\mu^2=p\times q$ such that $p$, $q$ are integers and there exists a $p$-order Hadamard or $(p+1)$-order Conference matrix. In particular, it is proved the maximum order of $\dot{\mathscr{B}}$ is $2s$ when either $q=1$, $s=c\mu^2=cp$ or $q=1$, $s=c(\mu^2+1)=c(p+1)$, $c=1,2,3,\cdots$. Futhermore, some extremal graphs are characterized.