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

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

A family of mixed graphs with large order and diameter 2

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order. We present a construction that provides mixed graphs of undirected degree qq, directed degree View the MathML sourceq-12 and order 2q22q2, for qq being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to View the MathML source9q2-4q+34 the defect of these mixed graphs is View the MathML source(q-22)2-14. In particular we obtain a known mixed Moore graph of order 1818, undirected degree 33 and directed degree 11 called Bosák’s graph and a new mixed graph of order 5050, undirected degree 55 and directed degree 22, which is proved to be optimal.Peer ReviewedPostprint (author's final draft

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft