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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.
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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 eigenvalue of the adjacency matrix of $S$ with $n$ vertices, $k=1,2,ldots,n$. Then the iota energy of $S$ is defined by $E(S)=sum_{k=1}^n|text{Im}(z_k)|$, where $text{Im}(z_k)$ is the imaginary part of eigenvalue $z_k$. In this paper, we consider a special graph class for bicyclic signed digraphs $mathcal{S}_n$ with $n$ vertices which have two vertex-disjoint signed directed even cycles. We give two iota energy orderings of bicyclic signed digraphs, one is including two positive or two negative directed even cycles, the other is including one positive and one negative directed even cycles.
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 $A(Gamma(G))$ of $Gamma(G)$. The random signed graph model $mathcal{G}_n(p, q)$ is defined as follows: Let $p, q ge 0$ be fixed, $0 le p+q le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.
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 elements of the spectrum of a finite digraph, which in this paper is allowed to have loops and parallel edges. We consider several different digraph spectra that have been studied in the literature, answering this question for the Laplace and adjacency spectra, their skew counterparts, the symmetric adjacency spectrum, the adjacency spectrum of the line digraph, the Hermitian adjacency spectrum, and the normalized Laplacian, considering in most cases two ways that these spectra can be defined in the presence of parallel edges. We show that the adjacency spectra of the digraph and line digraph are preserved by a subset of the moves, and the skew adjacency and Laplace spectra are preserved by the Cuntz splice. We give counterexamples to show that the other spectra are not preserved by the remaining moves. The same results hold if one restricts to the class of strongly connected digraphs.
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 prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.
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 networks where the nodes have the same type. Signed bipartite networks are different from classical signed networks, which contain two different node sets and signed links between two node sets. Signed bipartite networks can be commonly found in many fields including business, politics, and academics, but have been less studied. In this work, we firstly define the signed relationship of the same set of nodes and provide a new perspective for analyzing signed bipartite networks. Then we do some comprehensive analysis of balance theory from two perspectives on several real-world datasets. Specifically, in the peer review dataset, we find that the ratio of balanced isomorphism in signed bipartite networks increased after rebuttal phases. Guided by these two perspectives, we propose a novel Signed Bipartite Graph Neural Networks (SBGNNs) to learn node embeddings for signed bipartite networks. SBGNNs follow most GNNs message-passing scheme, but we design new message functions, aggregation functions, and update functions for signed bipartite networks. We validate the effectiveness of our model on four real-world datasets on Link Sign Prediction task, which is the main machine learning task for signed networks. Experimental results show that our SBGNN model achieves significant improvement compared with strong baseline methods, including feature-based methods and network embedding methods.
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