On (4,2)-digraph Containing a Cycle of Length 2

A diregular digraph is a digraph with the in-degree and out-degree of all vertices is constant. The Moore bound for a diregular digraph of degree d and diameter k is M_{d,k}=l+d+d^2+...+d^k. It is well known that diregular digraphs of order M_{d,k}, degree d>l tnd diameter k>l do not exist . A (d,k) -digraph is a diregular digraph of degree d>1, diameter k>1, and number of vertices one less than the Moore bound. For degrees d=2 and 3,it has been shown that for diameter k >= 3 there are no such (d,k)-digraphs. However for diameter 2, it is known that (d,2)-digraphs do exist for any degree d. The line digraph of K_{d+1} is one example of such (42)-digraphs. Furthermore, the recent study showed that there are three non-isomorphic(2,2)-digraphs and exactly one non-isomorphic (3,2)-digraph. In this paper, we shall study (4,2)-digraphs. We show that if (4,2)-digraph G contains a cycle of length 2 then G must be the line digraph of a complete digraph K_5

Degree/diameter problem for mixed graphs

The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs

Subdigraphs of Almost Moore Digraphs Induced by Fixpoints of an Automorphism

The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M(d,k)=1+d+d^2+...+d^k$ and almost Moore digraphs are digraphs with maximum out-degree d, diameter k and order M(d,k)-1. In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism varphi. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregular k-geodetic digraph of degree d'<d-1, order M(d',k)+1 and diameter k+1. As it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d=4 and d=5, where we prove that except in some special cases, all vertices will have the same order

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

Quantum Hall Ground States, Binary Invariants, and Regular Graphs

Extracting meaningful physical information out of a many-body wavefunction is often impractical. The polynomial nature of fractional quantum Hall (FQH) wavefunctions, however, provides a rare opportunity for a study by virtue of ground states alone. In this article, we investigate the general properties of FQH ground state polynomials. It turns out that the data carried by an FQH ground state can be essentially that of a (small) directed graph/matrix. We establish a correspondence between FQH ground states, binary invariants and regular graphs and briefly introduce all the necessary concepts. Utilizing methods from invariant theory and graph theory, we will then take a fresh look on physical properties of interest, e.g. squeezing properties, clustering properties, etc. Our methodology allows us to `unify' almost all of the previously constructed FQH ground states in the literature as special cases of a graph-based class of model FQH ground states, which we call \emph{accordion} model FQH states

Properties of locally checkable vertex partitioning problems in digraphs

While, for undirected graphs, locally checkable vertex subset and partitioning problems have been studied extensively, the equivalent directed problems have not received nearly as much attention yet. We take a closer look at the relationship between undirected and directed problems considering hardness. We extend some properties that have already been shown for undirected graphs to directed graphs. Furthermore, we explore some of the trivialities in directed problem definitions that do not appear in undirected ones. And finally, we construct and visualize digraph coverings to achieve a deeper understanding of their structure.Masteroppgave i informatikkINF399KMAMN-IN

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

AbstractFor a nonnegative n×n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A)=n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n×n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions