Line Digraph Research Articles

Super Connectivity of Line Graphs and Digraphs

The h-super connectivity κ h and the h-super edge-connectivity λ h are more refined network reliability indices than the connectivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1(L) = 2λ1(D), and that for a connected graph G and its line graph L, if one of κ1(L) and λ2(G) exists, then κ1(L) = λ2(G). This paper determines that κ1(B(d, n)) is equal to 4d -8 for n = 2 and d ≥ 4, and to 4d -4 for n ≥ 3 and d ≥ 3, and that κ1(K(d, n)) is equal to 4d - 4 for d ≥ 2 and n ≥ 2 except K(2, 2). It then follows that B(d, n) and K(d, n) are both super connected for any d ≥ 2 and n ≥ 1.

On endo-Cayley digraphs: The hamiltonian property

Given a finite abelian group A, a subset Δ ⊆ A and an endomorphism φ of A, the endo-Cayley digraph G A ( φ , Δ ) is defined by taking A as the vertex set and making every vertex x adjacent to the vertices φ ( x ) + a with a ∈ Δ . When A is cyclic and the set Δ is of the form Δ = { e , e + h , … , e + ( d - 1 ) h } , the digraph G is called a consecutive digraph. In this paper we study the hamiltonicity of endo-Cayley digraphs by using three approaches based on: line digraph, merging cycles and a generalization of the factor group lemma. The results are applied to consecutive digraphs.

Hamiltonicity of digraphs for universal cycles of permutations

The digraphs P ( n , k ) have vertices corresponding to length k permutations of an n set and arcs corresponding to ( k + 1 ) permutations. Answering a question of Starling, Klerlein, Kier and Carr we show that these digraphs are Hamiltonian for k ≤ n − 3 . We do this using restricted Eulerian cycles and the fact that P ( n , k ) is nearly the line digraph of P ( n , k − 1 ) . We also show that the digraphs P ( n , n − 2 ) are not Hamiltonian for n ≥ 4 using a result of Rankin on Cayley digraphs.

Independence number of iterated line digraphs

In this paper, we deal with the independence number of iterated line digraphs of a regular digraph G. We give pertinent lower bounds and give an asymptotic estimation of the ratio of the number of vertices of a largest independent set of the n th iterated line digraph of G.

Diameter vulnerability of iterated line digraphs in terms of the girth

Iterated line digraphs arise naturally in designing fault tolerant systems. Diameter vulnerability measures the increase in diameter of a digraph when some of its vertices or arcs fail. Thus, the study of diameter vulnerability is a suitable approach to the fault tolerance of a network. In this article we present some upper bounds for diameter vulnerability of iterated line digraphs LkG. Our bounds depend basically on the girth of the digraph G and on the number of iterations k. These bounds generalize some previous results on diameter vulnerability of line digraphs. Also, we apply our results to several important families of line digraphs such as Kautz digraphs and deBruijn generalized cycles, which contain deBruijn digraphs, the Reddy-Pradhan-Kuhl digraphs, and the butterflies. Our bounds allow us to obtain improvements in known results on diameter vulnerability for all these families. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 49-54 2005

On Arc-Regular Permutation Groups Using Latin Squares

For a given a permutation group G, the problem of determining which regular digraphs admit G as an arc-regular group of automorphism is considered. Groups which admit such a representation can be characterized in terms of generating sets satisfying certain properties, and a procedure to manufacture such groups is presented. The technique is based on constructing appropriate factorizations of (smaller) regular line digraphs by means of Latin squares. Using this approach, all possible representations of transitive groups of degree up to seven as arc-regular groups of digraphs of some degree is presented.

TRANSMISSION FAULT-TOLERANCE OF ITERATED LINE DIGRAPHS

The main result of this paper states that, if every cyclic modification of a d-regular digraph has super line-connectivity d, then the line digraph also has super line-connectivity d. Since many well-known interconnection network topologies, such as the Kautz digraphs, the de Bruijn digraphs, etc., can be constructed by iterating the line digraph construction, our result leads to several known and new connectivity results for these topologies, as shown later in the paper.

A characterization problem on underlying graphs of line digraphs

The underlying graph G ( D ) of a digraph D is arising when directions of edges are ignored. Chvátal and Ebenegger prove that recognising underlying graphs of line digraphs is an NP-complete problem. We note first that Chvátal and Ebengger's proof implies even more, namely that recognising underlying graphs of line digraphs of digraph in which each vertex has in-degree or out-degree at most one is an NP-complete problem. We then give a description of those underlying graphs which satisfy a supplementary condition on vertices of degree greater than two.

Diameter, short paths and superconnectivity in digraphs

A connected digraph is said to be superconnected if it is maximally connected and every minimum disconnecting set F consists of the vertices adjacent to or from a given vertex not belonging to F. Let δ be the minimum degree of the digraph and π be a positive integer such that π ⩽ ⌊ δ / 2 ⌋ when δ ⩾ 7 , or π ⩽ ⌊ ( δ - 2 ) / 2 ⌋ for δ ⩾ 5 . We prove that G is maximally connected or has a good superconnectivity if the diameter D ⩽ 2 ℓ π - 2 and ℓ 0 ⩾ 2 , where ℓ π is a generalization of the semigirth ℓ 0 introduced by Fàbrega and Fiol (J. Graph Theory 13(6) (1989) 657). We also show that G is maximally connected if π ⩽ ⌊ ( δ - 1 ) / 2 ⌋ and 3 ⩽ δ ⩽ 6 . In the edge case, it is enough that D ⩽ 2 ℓ π - 1 . Finally, the obtained results are applied to the iterated line digraphs.

Algebraic properties of a digraph and its line digraph

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length.

Functions and line digraphs

AbstractConsider two maps f and g from a set E into a set F such that f(x) ≠ g(x) for every x in E. Suppose that there exists a positive integer n such that for any element z in F either f−1(z) or g−1(z) has at most n elements. Then, E can be partitioned into 2n + 1 subsets E1, E2,…,E2n + 1 such that f(Ei)∩ g(Ei) = ϕ, 1 ≤ i ≤ 2n + 1. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 296–303, 2003

Weakly distance-regular digraphs

We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented.

Diameter vulnerability of GC graphs

Concern over fault tolerance in the design of interconnection networks has stimulated interest in finding large graphs with maximum degree Δ and diameter D such that the subgraphs obtained by deleting any set of s vertices have diameter at most D′, this value being close to D or even equal to it. This is the so-called ( Δ, D, D′, s)-problem. The purpose of this work has been to study this problem for s=1 on some families of generalized compound graphs. These graphs were designed by Gómez (Ars Combin. 29-B (1990) 33) as a contribution to the ( Δ, D)-problem, that is, to the construction of graphs having maximum degree Δ, diameter D and an order large enough. When approaching the mentioned problem in these graphs, we realized that each of them could be redefined as a compound graph, the main graph being the underlying graph of a certain iterated line digraph. In fact, this new characterization has been the key point to prove in a suitable way that the graphs belonging to these families are solutions to the ( Δ, D, D+1,1)-problem.

Heredity of the index of convergence of the line digraph

Let D be a digraph, and let L n ( D) and k( D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k( L n ( D))= k( D)=0 if every connected component of D is a directed cycle. (2) k( L n ( D))= k( D)+ n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k( D)⩽ k( L( D))⩽ k( D)+1.

On Moore bipartite digraphs

AbstractIn the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore‐like bound in terms of its diameter k and the maximum out‐degrees (d1, d2) of its partite sets of vertices. It has been proved that, when d1d2 > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003

Super link-connectivity of iterated line digraphs

Many interconnection networks can be constructed with line digraph iterations. A digraph has super link-connectivity d if it has link-connectivity d and every link-cut of cardinality d consists of either all out-links coming from a node, or all in-links ending at a node, excluding loop. In this paper, we show that the link-digraph iteration preserves super link-connectivity.

On the Digraph of a Unitary Matrix

Given a matrix M of size n, the digraph D on n vertices is said to be the digraph ofM , when $M_{ij}\neq 0$ if and only if (v,sub>i,v,sub>j) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph $\overrightarrow{L}D$ is the pattern of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and $\overrightarrow{L}D$ is the digraph of a unitary matrix, then $\overrightarrow{L}D$ is Hamiltonian. We observe that strong quadrangularity is sufficient to show that disconnected strongly regular graphs are the digraphs of unitary matrices and that n-paths, n-paths with loops at each vertex, n-cycles, directed trees, and trees are not.

Embedding iterated line digraphs in books

AbstractIn this paper, we present an upper bound on the pagenumber of an iterated line digraph Lk(G) of a digraph G. Our bound depends only on the digraph G and is independent of the number of iterations k. In particular, it is proved that the pagenumber of Lk(G) does not increase with the number of iterations k. This result generalizes previous results on book‐embeddings of some particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, and butterfly networks. Also, we apply our result to wrapped butterfly networks. © 2002 Wiley Periodicals, Inc.

Highly Arc-Transitive Digraphs With No Homomorphism Onto Z

In an infinite digraph D, an edge e' is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e'. Reachability is an equivalence relation and if D is 1-arc-transitive, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs. In Combinatorica, 13 (1993), Cameron, Praeger and Wormald asked if there exist highly arc-transitive digraphs (apart from directed cycles) for which the reachability relation is not universal and which do not have a homomorphism onto the two-way infinite directed path (a Cayley digraph of Z with respect to one generator). In view of an earlier result of Praeger in Australas. J. Combin., 3 (1991), such digraphs are either locally infinite or have equal in- and out-degree. In European J. Combin., 18 (1997), Evans gave an affirmative answer by constructing a locally infinite example. For each odd integer n >= 3, a construction of a highly arc-transitive digraph without property Z satisfying the additional properties that its in- and out-degree are equal to 2 and that the reachability equivalence classes induce alternating cycles of length 2n, is given. Furthermore, using the line digraph operator, digraphs having the above properties but with alternating cycles of length 4 are obtained.

Optimal design of diagnosable systems on networks constructed by graph operations

AbstractIn this paper, we consider self‐diagnosable systems on computer networks, known as one‐step t‐fault diagnosable systems introduced by Preparata and colleagues. In order to give network topologies, some graph operations such as the Cartesian product, the Kronecker product, and the line digraph operation are utilized. We study the diagnosability of networks constructed by the graph operations based on the idea of “highly structured system,” and then present some optimal networks. © 2002 Scripta Technica, Electron Comm Jpn Pt 3, 85(5): 1–9, 2002