Related papers: Structural Equivalence in Graphs and Complete Skel…

Let $\Gamma$ denote a finite, connected graph with vertex set $X$. Fix $x \in X$ and let $\varepsilon \ge 3$ denote the eccentricity of $x$. For mutually distinct scalars $\{\theta^*_i\}_{i=0}^\varepsilon$ define a diagonal matrix…

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $\Gamma(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article,…

We characterize connected tetravalent graphs $\Gamma$ which admit groups $M<H$ of automorphisms such that $\Gamma$ is $M$-half-arc-transitive and $H$-arc-transitive. Examples for each case are constructed, including a counter-example to a…

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma(v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and for any field $F$, let…

Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…

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…

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ is determined by its spectrum if every graph cospectral to it…

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in…

There has been a great deal of attention recently to graphs whose vertex set is a group, defined using the group structure. (The commuting graph, where two elements are joined if they commute, is the oldest and most famous example.) The…

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_\Gamma)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_\Gamma)$ has at most order…

The {\em power index} $\Theta(\Gamma)$ of a graph $\Gamma$ is the least order of a group $G$ such that $\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\em $\Gamma$-optimal} if $G$ has order $\Theta(\Gamma)$.…

In this short note we prove that a graph product $G_\Gamma$ of finitely generated abelian groups is semicomplete -- that is the kernel of the natural homomorphism ${\rm Aut}(G_\Gamma)\to{\rm Aut}(G_\Gamma^{ab})$ induced by the…

A regular equivalence between two graphs $\Gamma,\Gamma'$ is a pair of uniformly proper Lipschitz maps $V\Gamma\to V\Gamma'$ and $V\Gamma'\to V\Gamma$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence…

The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that…

Let $G_{g,b}$ be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface of genus $g$ with $b$ boundary components. We describe the set $A_{g,b}$ of all automorphisms of…

Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that…

The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants…

The Fibonacci cube $\Gamma_n$ is is the graph whose vertices are independent subsets of the path graph of length $n$, where two such vertices are considered adjacent if they differ by the addition or removal of a single element. Klav\v{z}ar…

Finding vertex-to-vertex correspondences in real-world graphs is a challenging task with applications in a wide variety of domains. Structural matching based on graphs connectivities has attracted considerable attention, while the…