相关论文: Orbit equivalence of graphs and isomorphism of gra…

The graph groupoids of directed graphs are topologically isomorphic if and only if there is a diagonal-preserving ring *-isomorphism between the Leavitt path algebras.

We study the notions of continuous orbit equivalence and eventual one-sided conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of row-finite higher-rank graphs with finitely many vertices and no sinks or sources. We…

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if…

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite…

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that…

We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite…

To determine that two given undirected graphs are isomorphic, we construct for them auxiliary graphs, using the breadth-first search. This makes capability to position vertices in each digraph with respect to each other. If the given graphs…

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph…

Two graphs having the same spectrum are said to be cospectral. Two graphs such that the absolute values of their nonzero eigenvalues coincide are singularly cospectral graphs. Cospectrality implies singular cospectrality, but the converse…

We prove two isomorphism-invariance theorems for groupoids associated with ultragraphs. These theorems characterize ultragraphs for which the topological full group of an associated groupoid is an isomorphism invariant. These results extend…

In this paper, we investigate certain graphs defined on groups, with a focus on infinite groups. The graphs discussed are the power graph, the enhanced power graph, and the commuting graph whose vertex set is a group $G$. The power graph is…

The paper proves two theorems concerning the set of periods of periodic orbits for maps of graphs that are homotopic to the constant map and such that the vertices form a periodic orbit. The first result is that if $v$ is not a divisor of…

A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…

In this article, we show that the stabilized automorphism group of free exact odometers arising from actions of finitely generated residually finite groups coincides with the topological full group of the odometer acting on itself by right…

We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).

An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…

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 show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate…

Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices…