12,498 research outputs found
Moore mixed graphs from Cayley graphs
A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in- and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M(r, z, k) for mixed graphs. When the order of G is close to M(r, z, k) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z, 3)-mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.This research has been partially supported by AGAUR from the Catalan Government under project
2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RBI00.Peer ReviewedPostprint (published version
Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite, and related graphs
We show that the infinitesimal generator of the symmetric simple exclusion
process, recast as a quantum spin-1/2 ferromagnetic Heisenberg model, can be
solved by elementary techniques on the complete, complete bipartite, and
related multipartite graphs. Some of the resulting infinitesimal generators are
formally identical to homogeneous as well as mixed higher spins models. The
degeneracies of the eigenspectra are described in detail, and the
Clebsch-Gordan machinery needed to deal with arbitrary spin-s representations
of the SU(2) is briefly developed. We mention in passing how our results fit
within the related questions of a ferromagnetic ordering of energy levels and a
conjecture according to which the spectral gaps of the random walk and the
interchange process on finite simple graphs must be equal.Comment: Final version as published, 19 pages, 4 figures, 40 references given
in full forma
Dense subgraph mining with a mixed graph model
In this paper we introduce a graph clustering method based on
dense bipartite subgraph mining. The method applies a mixed
graph model (both standard and bipartite) in a three-phase
algorithm. First a seed mining method is applied to find seeds
of clusters, the second phase consists of refining the seeds,
and in the third phase vertices outside the seeds are clustered.
The method is able to detect overlapping clusters, can handle
outliers and applicable without restrictions on the degrees of
vertices or the size of the clusters. The running time of the
method is polynomial. A theoretical result is introduced on
density bounds of bipartite subgraphs with size and local
density conditions. Test results on artificial datasets and
social interaction graphs are also presented
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