Spectra and energy of bipartite signed digraphs

Abstract: The set of distinct eigenvalues of a signed digraph $S$ together with their multiplicities is called its spectrum. The energy of a signed digraph $S$ with eigenvalues $z_1,z_2,\cdots,z_n$ is defined as $E(S)=\sum_{j=1}^{n}|\Re z_j|$, where $\Re z_j $ denotes real part of complex number $z_j$. In this paper, we show that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle of length $\equiv 0\pmod 4$ negative and each cycle of length $\equiv 2\pmod 4$ positive is of the form \ $$\phi_S(z)=z^n+\sum\limits_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}(-1)^j c_{2j}(S)z^{n-2j},$$\ where $c_{2j}(S)$ are nonnegative integers. We define a quasi-order relation in this case and show energy is increasing. It is shown that the characteristic polynomial of a bipartite signed digraph of order $n$ with each cycle negative has the form $$\phi_S(z)=z^n+\sum\limits_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}c_{2j}(S)z^{n-2j},$$ where $c_{2j}(S)$ are nonnegative integers. We study integral, real, Gaussian signed digraphs and quasi-cospectral digraphs and show for each positive integer $n\ge 4$ there exists a family of $n$ cospectral, non symmetric, strongly connected, integral, real, Gaussian signed digraphs (non cycle balanced) and quasi-cospectral digraphs of order $4^n$. We obtain a new family of pairs of equienergetic strongly connected signed digraphs and answer to open problem $(2)$ posed in Pirzada and Mushtaq, Energy of signed digraphs, Discrete Applied Mathematics 169 (2014) 195-205.