Related papers: On generalised Paley graphs and their automorphism…
We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every…
Let G be a finite group with identity e and H \neq \{e\} be a subgroup of G. The generalized non-coprime graph GAmma_{G,H} of G with respect to H is the simple undirected graph with G - \{e \}\) as the vertex set and two distinct vertices a…
A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular…
Let $G$ be an undirected graph of order $n$ and let $C_i$ be an $i$-cycle graph. $G$ is called pancyclic if $G$ contains a $C_i$ for any $i\in \{3,4,\ldots,n\}$. We show that the pancyclicity of specific Cayley graphs and the Cartesian…
Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps…
It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in…
In a recent paper [3], the authors introduced a map $\mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $\mathbb{F}_1$) with any given graph $\Gamma$. By base extension, a scheme…
A new family of strongly regular graphs, called the general symplectic graphs $Sp(2\nu, q)$, associated with nonsingular alternate matrices is introduced. Their parameters as strongly regular graphs, their chromatic numbers as well as their…
In this paper, we present constructions of abelian Paley type group schemes by using multiplicative characters of finite fields and Arasu-Dillon-Player difference sets. The constructions produce many new Paley type group schemes that were…
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…
A finite graph $\G$ is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of…
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from…
We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.
A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian…
A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs.…
We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the…
In this paper we characterize "large" regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that "large" association schemes become $P$-polynomial…
The isomorphism problem for digraphs is a fundamental problem in graph theory. This problem for Cayley digraphs has been extensively investigated over the last half a century. In this paper, we consider this problem for $m$-Cayley digraphs…
We describe a technique to determine the automorphism group of a geometrically represented graph, by understanding the structure of the induced action on all geometric representations. Using this, we characterize automorphism groups of…
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…