Related papers: Cayley graphs and symmetric interconnection networ…
Graphs defined over a finite ring are well-studied in the literature. Due to their nature, these types of graphs connect several branches of mathematics, including algebra, number theory, matrix theory, and representation theory. In recent…
In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph $X$ in the group of symmetries of the Jacobian of $X$. As a consequence we show that if a $3$-edge-connected graph…
A digraph X=(V, E) is max-\lambda, if \lambda(X)=\delta(X). A digraph X is super-\lambda if every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all…
In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is…
This paper is the second in a series of studies on developing efficient artificial intelligence-based approaches to pathfinding on extremely large graphs (e.g. $10^{70}$ nodes) with a focus on Cayley graphs and mathematical applications.…
Connectivity is a central notion of graph theory and plays an important role in graph algorithm design and applications. With emerging new applications in networks, a new type of graph connectivity problem has been getting more…
A relational structure is (connected-)homogeneous if every isomorphism between finite (connected) substructures extends to an automorphism of the structure. We investigate notions which generalise (connected-)homogeneity, where…
Given a finite simple graph $\cG$ with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of $\cG$, interpreted as transpositions. We show that, if $\cG$ is complete multipartite, the…
Topological metrics of graphs provide a natural way to describe the prominent features of various types of networks. Graph metrics describe the structure and interplay of graph edges and have found applications in many scientific fields. In…
Motivated by very large-scale communication networks, we newly introduce exponentiation of graphs. Using the exponential operation on graphs, we can construct various graphs of multi-exponential order with logarithmic diameter. We show that…
When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More…
This article lays the foundations for an analogue of geometric group theory that studies actions on graphs by right quasigroups, including racks and quandles. We study markings of graphs that realize racks, and we introduce (di)graph…
Let $G$ be a group. The power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x,y$ are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence…
A graph is a powerful concept for representation of relations between pairs of entities. Data with underlying graph structure can be found across many disciplines and there is a natural desire for understanding such data better. Deep…
It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of…
A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition…
As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…
We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all…
The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective…
We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to…