Related papers: Six signed Petersen graphs, and their automorphism…
A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement…
The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral…
We introduce the concept of deficiency in signed graphs. The deficiency of a coloration is the number of unused colors. We classify the deficiency of 2-chromatic graphs. There are four decision problems about the minimum and maximum…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult…
We characterize when the spectral variation of the signed Laplacian matrices is integral after a new edge is added to a signed graph. As an application, for every fixed signed complete graph, we fully characterize the class of signed graphs…
In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the…
A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
Let $G$ be a graph whose edges are each assigned one of the $m$-colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ with respect $\pi \in \Gamma$ permutes the colours of the edges…
Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their…
Three new graph invariants are introduced which may be measured from a quantum graph state and form examples of a framework under which other graph invariants can be constructed. Each invariant is based on distinguishing a different number…
We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those…
A strong edge-coloring of a graph is a proper edge-coloring, in which the edges of every path of length 3 receive distinct colors; in other words, every pair of edges at distance at most 2 must be colored differently. The least number of…
In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and…
This note provides an introduction to selected topics in algebraic graph theory, including strongly regular graphs, Steiner systems, and automorphism groups. We describe constructions and properties of notable graphs such as the Petersen…
We analyse signed networks from the perspective of balance theory which predicts structural balance as a global structure for signed social networks that represent groups of friends and enemies. The scarcity of balanced networks encouraged…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the…
We introduce the idea of an n-simplex graph and games upon simplicial complexes. We then define moves on a labeled graph and pose the problem of whether given two labelings of a graph it is possible to change one into another via these…