Line and subdivision graphs determined by T4-gain graphs

Let T4 = (±1, ±i) be the subgroup of fourth roots of unity inside T, the multiplicative group of complex units. For a T4-gain graph Φ = (Γ,T4, ϕ), we introduce gain functions on its line graph L(Γ) and on its subdivision graph S(Γ). The corresponding gain graphs L(Φ) and S(Φ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ, and the adjacency characteristic polynomials of L(Φ) and S(Φ). A suitably defined incidence matrix for T4-gain graphs plays an important role in this contex

The H-Line Signed Graph of a Signed Graph

For standard terminology and notion in graph theory we refer the reader to Harary; the non-standard will be given in this paper as and when required. We treat only finite simple graphs without self loops and isolates

Negation Switching Equivalence in Signed Graphs

Unless mentioned or defined otherwise, for all terminology and notion in graph theory the reader is refer to [8]. We consider only finite, simple graphs free from self-loops

On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy

In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor signed graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page

Characterization of Line-Consistent Signed Graphs

The line graph of a graph with signed edges carries vertex signs. A vertex-signed graph is consistent if every circle (cycle, circuit) has positive vertex-sign product. Acharya, Acharya, and Sinha recently characterized line-consistent signed graphs, i.e., edge-signed graphs whose line graphs, with the naturally induced vertex signature, are consistent. Their proof applies Hoede's relatively difficult characterization of consistent vertex-signed graphs. We give a simple proof that does not depend on Hoede's theorem as well as a structural description of line-consistent signed graphs.Comment: 5 pages. V2 defines sign of a walk and corrects statement of Theorem 4 ("is balanced and" was missing); also minor copyeditin

Total Minimal Dominating Signed Graph

Cartwright and Harary considered graphs in which vertices represent persons and the edges represent symmetric dyadic relations amongst persons each of which designated as being positive or negative according to whether the nature of the relationship is positive (friendly, like, etc.) or negative (hostile, dislike, etc.). Such a network S is called a signed graph. Signed graphs are much studied in literature because of their extensive use in modeling a variety socio-psychological process and also because of their interesting connections with many classical mathematical systems