This paper presents a spectral analysis of signed networks from both theoretical and practical aspects. On the theoretical aspect, we conduct theoretical study based on matrix perturbation theorem for analyzing community structures of complex signed networks and show how the negative edges affect distributions and patterns of node spectral coordinates in the spectral space. We prove and demonst...

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13–28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the rôle of circuits is taken over by signed circuits of a signed graph which are known to occur in two types – either balanced circuits or pairs of disjoint unbalanced circuits connecte...

This is an expository survey of the uses of matrices in the theory of simple graphs with signed edges. A signed simple graph is a graph, without loops or parallel edges, in which every edge has been declared positive or negative. For many purposes the most significant thing about a signed graph is not the actual edge signs, but the sign of each circle (or ‘cycle’ or ’circuit’), which is the pro...

This paper is to introduce circuit, bond, flow, and tension spaces and lattices for signed graphs, and to study the relations among these spaces and lattices. The key ingredient is to introduce circuit and bond characteristic vectors so that the desired spaces and lattices can be defined such that their dimensions and ranks match well to that of matroids of signed graphs. The main results can b...

As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least 2k + 1 admits a homomorphism to PC2k = (Z 2 , {e1, e2, · · · , e2k, J}) where ei’s are standard basis and J is all 1 vector. Noting that PC2k itself is of odd-girth 2k + 1, in this work we show that if the conjecture is true, then PC2k is an optimal such a graph both with respect to number ...

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

We generalize three classical characterizations of line graphs to signed and gain graphs: the Krausz's characterization, van Rooij Wilf's characterization Beineke's characterization. In particular, we present a list forbidden subgraphs characterizing class gain-line graphs. case graph whose underlying is graph, this consists exactly four Under same hypothesis, prove that if only its eigenvalues...

I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (‘Laplacian’) matrices, line graphs, and very strong regularity.

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ∈ {3, 4, 5, 6}, every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of le...