Related papers: On monoid graphs
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on…
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…
We study the unitary Cayley graph of a matrix semiring. We find bounds for its diameter, clique number and independence number, and determine its girth. We also find the relationship between the diameter and the clique number of a unitary…
We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the…
In this paper, we show that every bipartite distance-regular Cayley graph with diameter $3$ can be constructed on the semidirect product of a group and $\mathbb{Z}_{2}$, except possibly for one case.
A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases…
Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems…
Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this…
In this paper, we study commutative zero-divisor semigroups determined by graphs. We prove a uniqueness theorem for a class of graphs. We show two classes of graphs that have no corresponding semigroups. In particular, any complete graph…
This is an expository paper which provides a quick introduction to Boolean inverse semigroups and their type monoids, with the emphasis on techniques and insights of the theory, and also treats the connection of the type monoid…
In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…
We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main…
The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal…
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…
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…
We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings'…
We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for…
We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive…
We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs $C$ is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs…
In this work, we explore edge direction, transitivity, and connectedness of Cayley graphs of gyrogroups. More specifically, we find conditions for a Cayley graph of a gyrogroup to be undirected, transitive, and connected. We also show a…