Symmetric Digraph Research Articles

On symmetric digraphs of the congruence [formula omitted

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , … , n − 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is symmetric of order M if its set of components can be partitioned into subsets of size M with each subset containing M isomorphic components. We generalize earlier theorems by Szalay, Carlip, and Mincheva on symmetric digraphs G ( n , 2 ) of order 2 to symmetric digraphs G ( n , k ) of order M when k ≥ 2 is arbitrary.

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, K n ∗ , exist if and only if n is odd with n ≠ 15 and n ≠ p α for p an odd prime and α ≥ 2 or n ≡ 2 ( mod 4 ) with n ≠ 2 p α for p an odd prime and α ≥ 1 . We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n / 2 vertex-independent digons, ( K n − I ) ∗ , exist if and only if n ≡ 0 ( mod 4 ) .

Exponents of 2-coloring of symmetric digraphs

A 2-coloring ( G 1 , G 2 ) of a digraph is 2-primitive if there exist nonnegative integers h and k with h + k > 0 such that for each ordered pair ( u , v ) of vertices there exists an ( h , k ) -walk in ( G 1 , G 2 ) from u to v. The exponent of ( G 1 , G 2 ) is the minimum value of h + k taken over all such h and k. In this paper, we consider 2-colorings of strongly connected symmetric digraphs with loops, establish necessary and sufficient conditions for these to be 2-primitive and determine an upper bound on their exponents. We also characterize the 2-colored digraphs that attain the upper bound and the exponent set for this family of digraphs on n vertices.

Large vertex symmetric digraphs

AbstractWe study the problem of finding large vertex symmetric digraphs with given degree Δ and diameter D. This problem is also called the (Δ,D)‐problem or the degree/diameter problem for vertex symmetric digraphs. It consists of finding vertex symmetric digraphs with the order as large as possible for a given degree and diameter. This paper proposes a new general family of large vertex symmetric digraphs. This new family has order and provides the best values for asymptotic values of the degree and diameter in general. Moreover, the presented family also provides a large number of new values in the table of the largest known vertex symmetric (Δ,D)‐digraphs. Furthermore, the new family is used to obtain new large vertex symmetric digraphs by means of known compounding techniques. Finally, the table of the largest known vertex symmetric digraphs is provided, where most of the entries correspond to the work presented in this article. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(4), 241–250 2007

Semiregular automorphisms of arc-transitive graphs with valency pq

It was conjectured (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81]) that every vertex-transitive digraph has a semiregular automorphism, that is, a nonidentity automorphism having all orbits of equal length. Despite several partial results supporting its content, the conjecture remains open. In this paper, it is shown that the conjecture holds whenever the graph is arc-transitive of valency p q , where p and q are primes ( p may equal q ), and such that its automorphism group has a nonabelian minimal normal subgroup with at least three orbits on the vertex set.

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 λ v . 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.

Locating Servers for Reliability and Affine Embeddings

Consider the problem of locating servers in a network for the purpose of storing data, performing an application, etc., so that at least one server will be available to clients even if up to $k$ component failures occur throughout the network. Letting $G = (V,E)$ be the graph with vertex set $V$ and edge set $E$ representing the topology of the network, and letting $L \subseteq V$ be a set of potential locations for the servers, a fundamental problem is to determine a minimum-size set $S \subseteq L$ such that the network remains connected to $S$ even if up to $k$ component failures occur throughout the network. We say that such a set $S$ is $k$-fault-tolerant. In this paper we present an algebraic characterization of $k$-fault-tolerant sets in terms of affine embeddings of $G$ in $k$-dimensional Euclidean space. Employing this characterization, we present a polynomial-time Monte Carlo algorithm for computing a minimum-size $k$-fault-tolerant subset $S$ of $L$. In fact, we solve the following more general problem for directed networks: given a digraph $G = (V,E)$ (an undirected graph is equivalent to a symmetric digraph) and a subset $L \subseteq V$, we find a $k$-fault-tolerant subset $S$ of $L$ having minimum cost, where a unary integer cost $c(v)$ is associated with locating a server at vertex $v \in V$.

Isomorphic factorization, the Kronecker product and the line digraph

In this paper, we investigate isomorphic factorizations of the Kronecker product graphs. Using these relations, it is shown that (1) the Kronecker product of the d-out-regular digraph and the complete symmetric digraph is factorized into the line digraph, (2) the Kronecker product of the Kautz digraph and the de Bruijn digraph is factorized into the Kautz digraph, (3) the Kronecker product of binary generalized de Bruijn digraphs is factorized into the binary generalized de Bruijn digraph.

Minimal normal subgroups of transitive permutation groups of square-free degree

It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to A 5 , A 7 , and PSL ( 2 , 29 ) . Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605–615]).

Semiregular automorphisms of vertex-transitive graphs of certain valencies

It is shown that a vertex-transitive graph of valency p + 1 , p a prime, admitting a transitive action of a { 2 , p } -group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605–615]).

A note on total colourings of digraphs

In the paper two different arc-colourings and two associated with the total colourings of digraphs are considered. In one of these colourings we show that the problem of calculating the total chromatic index reduces to that of calculating the chromatic number of the underlying graph. In the other colouring we find the total chromatic indices of complete symmetric digraphs and tournaments.

Divisibility conditions in almost Moore digraphs with selfrepeats

Abstract Moore digraph is a digraph with maximum out-degree d, diameter k and order M d , k = 1 + d + … + d k . Moore digraphs exist only in trivial cases if d = 1 (i.e., directed cycle C k ) or k = 1 (i.e., complete symmetric digraph). Almost Moore digraphs are digraphs of order one less than Moore bound. We shall present new properties of almost Moore digraphs with selfrepeats from which we prove nonexistence of almost Moore digraphs for some k and d.

Path Coloring on Binary Caterpillars

The path coloring problem is to assign the minimum number of colors to a given set of directed paths on a given symmetric digraph D so that no two paths sharing an arc have the same color. The problem has applications to efficient assignment of wavelengths to communications on WDM optical networks. In this paper, we show that the path coloring problem is NP-hard even if the underlying graph of D is restricted to a binary caterpillar. Moreover, we give a polynomial time algorithm which constructs, given a binary caterpillar G and a set of directed paths on the symmetric digraph associated with G, a path coloring of with at most ⌈8/5L⌉ colors, where L is the maximum number of paths sharing an edge. Furthermore, we show that no local greedy path coloring algorithm on caterpillars in general uses less than ⌈8/5L⌉ colors.

Connectedness of digraphs and graphs under constraints on the conditional diameter

Given a digraph G with minimum degree δ and an integer 0≤ ν ≤ δ, consider every pair of vertex subsets V1 and V2 such that both the minimum out-degree of the induced subdigraph G[V1] and the minimum in-degree of G[V2] are at least ν. The conditional diameter Dν of G is defined as the maximum of the distances d(V1, V2) between any two such vertex subsets. Clearly, D0 is the standard diameter and D0 ≥ D1 ≥ ··· ≥ Dδ holds. In this article, we guarantee appropriate lower bounds for the connectivities and superconnectivities of a digraph G when Dν ≤ h(ℓπ), h(ℓπ) being a function of the parameter ℓπ—which is related to the shortest paths in G. As a corollary of these results, we give some constraints of the kind Dν ≤ h(ℓπ), which assure that the digraph is maximally connected, maximally edge-connected, superconnected, or edge-superconnected, extending other previous results of the same kind. Similar statements can be obtained for a graph as a direct consequence of those for its associated symmetric digraph. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 80–87 2005

On the maximum even factor in weakly symmetric graphs

As a common generalization of matchings and matroid intersection, Cunningham and Geelen introduced the notion of path-matchings, then they introduced the more general notion of even factor in weakly symmetric digraphs. Here we give a min–max formula for the maximum cardinality of an even factor. Our proof is purely combinatorial. We also provide a Gallai–Edmonds-type structure theorem for even factors.

Bartholdi zeta functions of digraphs

We give a determinant expression for the Bartholdi zeta function of a digraph which is not symmetric. This is a generalization of Bartholdi’s result on the Bartholdi zeta function of a graph or a symmetric digraph.

Cycle decompositions IV: complete directed graphs and fixed length directed cycles

We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m, where 2⩽ m⩽ n.

Decomposition formulas of zeta functions of graphs and digraphs

We give a decomposition formula of the zeta function of a regular covering of a graph G with respect to equivalence classes of prime, reduced cycles of G. Furthermore, we give a decomposition formula of the zeta function of a g-cyclic Γ-cover of a symmetric digraph D with respect to equivalence classes of prime cycles of D, for any finite group Γ and g∈ Γ.

The Spectra of Cycle Prefix Digraphs

Cycle prefix digraphs comprise a class of vertex symmetric digraphs with many interesting properties, such as large order for a given degree and diameter, Hamiltonicity, and hierarchical structure. From their known structural properties, we determine the spectra of the digraphs. We also show that, although cycle prefix digraphs are not distance regular according to the usual definition, they have properties that fully characterize, in the undirected case, distance regular graphs.

Weighted zeta functions of digraphs

We define a weighted zeta function of a digraph and a weighted L-function of a symmetric digraph, and give determinant expressions of them. Furthermore, we give a decomposition formula for the weighted zeta function of a g-cyclic Γ-cover of a symmetric digraph for any finite group Γ and g∈ Γ. A decomposition formula for the weighted zeta function of an oriented line graph L → ( G ̃ ) of a regular covering G ̃ of a graph G is given. Furthermore, we define a weighted L-function of an oriented line graph L → (G) of G, and present a factorization formula for the weighted zeta function of L → ( G ̃ ) by weighted L-functions of L → (G) . As a corollay, we obtain a factorization formula for the multiedge zeta function of G ̃ given by Stark and Terras.