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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 $equiv 0pmod 4$ negative and each cycle of length $equiv 2pmod 4$ positive is of the form $$phi_S(z)=z^n+sumlimits_{j=1}^{lfloor{frac{n}{2}}rfloor}(-1)^j c_{2j}(S)z^{n-2j},$$ 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 $$phi_S(z)=z^n+sumlimits_{j=1}^{lfloor{frac{n}{2}}rfloor}c_{2j}(S)z^{n-2j},$$ where $c_{2j}(S)$ are nonnegative integers. We study integral, real, Gaussian signed digraphs and quasi-cospectral digraphs and show for each positive integer $nge 4$ 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.
The concept of energy of a signed digraph is extended to iota energy of a signed digraph. The energy of a signed digraph $S$ is defined by $E(S)=sum_{k=1}^n|text{Re}(z_k)|$, where $text{Re}(z_k)$ is the real part of eigenvalue $z_k$ and $z_k$ is the
A signed graph $Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Gamma(G)$. The energy of a signed graph $Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(G
Eilers et al. have recently completed the geometric classification of unital graph $C^ast$-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the nonzero eleme
We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we pro
Signed networks are such social networks having both positive and negative links. A lot of theories and algorithms have been developed to model such networks (e.g., balance theory). However, previous work mainly focuses on the unipartite signed netwo