Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.     

Multipartite Moore digraphs

We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a multipartite Moore digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a r-partite Moore digraph with interpartite outdegree δ > 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.     

Spectra of digraphs

In this primarily expository paper we survey classical and some more recent results on the spectra of digraphs, equivalently, the spectra of (0,1)-matrices, with emphasis on the spectral radius.     

Ihara zeta functions of digraphs

After defining and exploring some of the properties of Ihara zeta functions of digraphs, we improve upon Kotani and Sunada’s bounds on the poles of Ihara zeta functions of undirected graphs by considering digraphs whose adjacency matrices are directed edge matrices.     

Exixtence of vertices of local connectivityk in digraphs of large outdegree

For every positive integerk, there is a positive integerf(k) such that every finite digraph of minimum outdegreef(k) contains verticesx, y joined byk openly disjoint paths.     

Eigenvalues and colorings of digraphs

Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.     

Spectral radius of digraphs with given dichromatic number

Let D be a digraph with vertex set V(D). A partition of V(D) into k acyclic sets is called a k-coloring of D. The minimum integer k for which there exists a k-coloring of D is the dichromatic number χ(D) of the digraph D. Denote G_n,k the set of the digraphs of order n with the dichromatic number k≥2. In this note, we characterize the digraph which has the maximal spectral radius in G_n,k. Our result generalizes the result of [8] by Feng et al.     

On digraphs with no two disjoint directed cycles

We obtain a result on configurations in 2-connected digraphs with no two disjoint dicycles. We derive various consequences, for example a short proof of the characterization of the minimal digraphs having no vertex meeting all dicycles and a polynomially bounded algorithm for finding a dicycle through any pair of prescribed arcs in a digraph with no two disjoint dicycles, a problem which is NP-complete for digraphs in general.     

Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs

In [B.M. Kim, B.C. Song, W. Hwang, Primitive graphs with given exponents and minimum number of edges, Linear Algebra Appl. 420 (2007) 648-662], the minimum number of edges of a simple graph on n vertices with exponent k was determined. In this paper, we completely determine the minimum number, H(n,k), of arcs of primitive non-powerful symmetric loop-free signed digraphs on n vertices with base k, characterize the underlying digraphs which have H(n,k) arcs when k is 2, nearly characterize the case when k is 3 and propose an open problem.     

Longest paths in digraphs

In this paper, we give a sufficient condition on the degrees of the vertices of a digraph to insure the existence of a path of given length, and we characterize the extremal graphs.     

Disjoint cycles in digraphs

We show that, for each natural numberk, these exists a (smallest) natural numberf(k) such that any digraph of minimum outdegree at leastf(k) containsk disjoint cycles. We conjecture thatf(k)=2k−1 and verify this fork=2 and we show that, for eachk≧3, the determination off(k) is a finite problem. Dedicated to Paul Erdős on his seventieth birthday     

Bartholdi zeta and L-functions of weighted digraphs, their coverings and products

Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 19 (1966) 219-235], many kinds of zeta functions and L-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and L-functions of a graph contain the following: (1) defining a zeta function, (2) defining an L-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of L-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an L-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the L-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the L-functions.     

Arc-transitive digraphs with quasiprimitive local actions

Let Γ be a finite G-vertex-transitive digraph. The in-local action of $(\mathrm{\Gamma },G)$ is the permutation group $L_{?}$ induced by a vertex-stabiliser on the set of in-neighbours of the corresponding vertex. The out-local action$L_{+}$ is defined analogously. Note that $L_{?}$ and $L_{+}$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_{?},L_{+})$ are possible. We prove some general results, but pay special attention to the case when $L_{?}$ and $L_{+}$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.     

On a property ofn-edge-connected digraphs

It is proved that for every pair of verticesx, y in a finiten-edge-connected digraphD there is such a pathP fromx toy that the digraphD′ arising fromD by deleting the edges ofP is (n−1)-edge-connected.     

Packing circuits in eulerian digraphs

LetG be an eulerian digraph; let (G) be the maximum number of pairwise edge-disjoint directed circuits ofG, and (G) the smallest size of a set of edges that meets all directed circuits ofG. Borobia, Nutov and Penn showed that (G) need not be equal to (G). We show that (G)=(G) provided thatG has a linkless embedding in 3-space, or equivalently, if no minor ofG can be converted toK ₆ by –Y andY– operations.     

The spectra of some families of digraphs

Let G be a digraph (or a graph, when seen as a symmetric digraph) with adjacency matrix A, having the eigenvalue λ with associated eigenvector v. As it is well known, the entries of v can be interpreted as charges in each vertex. Then, the linear transformation v ? Av corresponds to a natural displacement of charges, where each vertex sends a copy of its charge to its in-neighbors and absorbs a copy of the charges of its out-neighbors, so the resulting charge distribution is just λv. In this work we use this approach to derive some old and new results about the spectral characterization of G. More precisely, we show how to obtain the spectra of some families of (di)graphs, such as the partial line digraphs and the line graphs of regular or semiregular graphs.     

Highly connected non-2-linked digraphs

For every natural numberk there exists a stronglyk-connected digraph which contains two vertices through which there is no directed cycle.     

The smallest eigenvalue of the signless Laplacian

Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless Laplacian.     

Degree sets for digraphs

The degree setD _D of a digraphD is the set of outdegrees of the vertices ofD. For a finite, nonempty setS of nonnegative integers, it is shown that there exists an asymmetric digraph (oriented graph)D such thatD _D =S. Furthermore, the minimum order of such a digraphD is determined. Also, given two finite sequences of nonnegative integers, a necessary and sufficient condition is provided for which these sequences are the outdegree sequences of the two sets of an asymmetric bipartite digraph.     

The smallest signless Laplacian spectral radius of graphs with a given clique number

The signless Laplacian spectral radius of a graph G is the largest eigenvalue of its signless Laplacian matrix. In this paper, the first four smallest values of the signless Laplacian spectral radius among all connected graphs with maximum clique of size greater than or equal to 2 are obtained.