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The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyn's and Temperley--Fisher's ideas about spectral properties of weighted adjacency matrices of planar…
The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a…
Signed graphs are an emergent way of representing data in a variety of contexts where antagonistic interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability for…
A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the…
The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. This matrix is first defined in 2018 by Wang et al. \cite{1}. In this…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…
Graph generation is integral to various engineering and scientific disciplines. Nevertheless, existing methodologies tend to overlook the generation of edge attributes. However, we identify critical applications where edge attributes are…
The vertex-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed…
The edge-degeneracy model is an exponential random graph model that uses the graph degeneracy, a measure of the graph's connection density, and number of edges in a graph as its sufficient statistics. We show this model is relatively…
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…
Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…
Two signed graphs are called switching isomorphic if one of them is isomorphic to a switching equivalent of the other. To determine the number of switching non-isomorphic signed graphs on a specific graph, we will establish a method based…
An oriented graph $G^\sigma$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^\sigma$. Let $S(G^\sigma)$ denote the skew-adjacency matrix of $G^\sigma$. The rank of the skew-adjacency matrix of…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
The number of the positive, negative and zero eigenvalues in the spectrum of the (edge)-weighted graph $G$ are called positive inertia index, negative inertia index and nullity of the weighted graph $G$, and denoted by $i_+(G)$, $i_-(G)$,…
Social and information networks are gaining huge popularity recently due to their various applications. Knowledge representation through graphs in the form of nodes and edges should preserve as many characteristics of the original data as…
Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific…
Let $G$ be a simple graph with an orientation $\sigma$, which assigns to each edge a direction so that $G^\sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^\sigma$. In this paper, we define a…
In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of $n \times n$ involutions with spectrum equal to $\{ \lambda_1^{n-2}, \lambda_2^{2} \}$ for some $\lambda_1$ and some…