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

On Total Regularity of Mixed Graphs with Order Close to the Moore Bound

The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter $k$, maximum undirected degree $\leq r$ and maximum directed out-degree $\leq z$. Similarly one can search for the smallest possible $k$-geodetic mixed graphs with minimum undirected degree $\geq r$ and minimum directed out-degree $\geq z$. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for $k = 2$ and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For $k = 2$, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of Lopez and Miret. We also present partial results for larger $k$. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one

A revised Moore bound for mixed graphs

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of vertices of such graphs. We show that for diameters larger than three, this bound can be reduced and we present a corrected Moore bound for mixed graphs, valid for all diameters and for all combinations of undirected and directed degrees

On mixed radial Moore graphs of diameter 3

Radial Moore graphs and digraphs are extremal graphs related to the Moore ones where the distance-preserving spanning tree is preserved for some vertices. This leads to classify them according to their proximity to being a Moore graph or digraph. In this paper we deal with mixed radial Moore graphs, where the mixed setting allows edges and arcs as different elements. An exhaustive computer search shows the top ranked graphs for an specific set of parameters. Moreover, we study the problem of their existence by providing two infinite families for different values of the degrees and diameter $3$. One of these families turns out to be optimal

An overview of the degree/diameter problem for directed, undirected and mixed graphs

A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems.Peer Reviewe

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs