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Cyclically ordered graphs, or cogs, sit between abstract graphs and cellularly embedded graphs. They arise naturally in topological graph theory, knot theory, and mathematical biology. We develop a formal theory of cogs and establish a…

We show that every bridgeless cubic graph $G$ on $n$ vertices other than the Petersen graph has a 2-factor with at most $2(n-2)/15$ circuits of length $5$. An infinite family of graphs attains this bound. We also show that $G$ has a…

Circulant graphs are a widely studied family of graphs whose members possess varying amounts of symmetry. Although considerable progress has been made in finding the automorphism groups of circulant graphs under certain restrictions, a…

Define an embedding of graph $G=(V,E)$ with $V$ a finite set of distinct points on the unit circle and $E$ the set of line segments connecting the points. Let $V_1,\ldots,V_k$ be a labeled partition of $V$ into equal parts. A 2-factor is…

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ…

A graph $\G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over $H$. Such a graph $\G$ is called {\em normal\/} if $H$ is normal in the full…

A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the…

A {\em cyclic graph} is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model…

A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists…

This document begins by reviewing recent progress that has been made by taking a combinatorial perspective on the $c_2$ invariant, an arithmetic graph invariant with connections to Feynman integrals. Then it proceeds to report on some…

Let $\Gamma$ be a multigraph with for each vertex a cyclic order of the edges incident with it. For $n \geq 3$, let $D_{2n}$ be the dihedral group of order $2n$. Define $\mathbb{D} := \{(\begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix})…

A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected…

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their…

We give a complete characterisation of the cubic graphs with no eigenvalues in the interval $(-2,0)$. There is one thin infinite family consisting of a single graph on $6n$ vertices for each $n \geqslant 2$, and five ``sporadic'' graphs,…

Based on a recent development in the area of error control coding, we introduce the notion of convolutional factor graphs (CFGs) as a new class of probabilistic graphical models. In this context, the conventional factor graphs are referred…

Quantum symmetry of graph $C^{*}$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots…

Let $G$ be a finite group, and $\alpha$ a nontrivial character of $G$. The McKay graph $\mathcal{M}(G,\alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $\chi_1$ to $\chi_2$ if $\chi_2$ is a constituent of…

Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…

A finite graph $\Gamma$ is called $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. We study a family of symmetric graphs, called the unitary…