Equivalence relations on the edge set of a graph G that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of G, which in turn give rise to quotient graphs that can have a rich product structure even if G itself is prime.

We consider a free action of an Ore semigroup on a higher-rank graph, and the induced action by endomorphisms of the C ∗-algebra of the graph. We show that the crossed product by this action is stably isomorphic to the C ∗-algebra of a quotient graph. Our main tool is Laca’s dilation theory for endomorphic actions of Ore semigroups on C ∗-algebras, which embeds such an action in an automorphic ...

This extended abstract presents a submission to the Music Information Retrieval Evaluation eXchange (MIREX) in the Query-by-Humming/Singing task. Melodies are represented by quotiented sequences. A quotiented sequence is a sequence graph defined with an additional equivalent relation on its vertices and such that the quotient graph is also a sequence graph. The core of the method relies on an a...

A graph is symmetric or arc-transitive if its automorphism group acts transitively on vertices, edges and arcs. Let p, q be odd primes with p, q ≥ 5 and X a q-valent symmetric graph of order 4p. In this paper, we proved that X K4p with 4p-1=q, X K2p,2p-2pK2 with 2p-1=q, the quotient graph of X is isomorphic to Kp,p and p=q, or K2p and 2p-1=q.

We study minimal free resolutions of edge ideals of bipartite graphs. We associate a directed graph to a bipartite graph whose edge ideal is unmixed, and give expressions for the regularity and the depth of the edge ideal in terms of invariants of the directed graph. For some classes of unmixed edge ideals, we show that the arithmetic rank of the ideal equals projective dimension of its quotient.

This paper examines the relationship between the symplectic quotientX//G of a Hamiltonian G-manifold X , and the associated symplectic quotient X//T , where T ⊂ G is a maximal torus, in the case in which X//G is a compact manifold or orbifold. The three main results are: a formula expressing the rational cohomology ring of X//G in terms of the rational cohomology ring of X//T ; an ‘integration’...

Throughout this paper the actions of groups on graphs with inversions are allowed. An element g of a group G is called inverter if there exists a tree X where G acts such that g transfers an edge of X into its inverse. A group G is called accessible if G is finitely generated and there exists a tree on which G acts such that each edge group is finite, no vertex is stabilized by G, and each vert...

The dilation coefficient of a graph representation is defined as the quotient of the longest and the shortest edge representation. The minimum of the dilation coefficients over all planar representations of a graph G is called the dilation coefficient of the graph G. The dilation coefficient of different planar representations of complete graphs is considered and upper and lower bounds for the ...

Let (B,MB) denote a regular local ring of dimension two. Let H be a finite cyclic group acting on B, and let Z := Spec(B). Assume that the action of H on Spec(B) is free off the closed point, and thatMB is the only singular point of Z. Let f : X → Z be a resolution of the singularity, minimal with the property that the irreducible components of f(MB ) are smooth with normal crossings. Such a re...