Iterated Line Graphs Research Articles

Minimum Number of Components of 2-Factors in Iterated Line Graphs

It is well known that it is NP-hard to determine the minimum number of components of a 2-factor in a graph, even for iterated line graphs. In this paper, we determine the minimum number of components of 2-factors in iterated line graphs of some special tree-like graphs. It extends some known results.

Seidel energy of iterated line graphs of regular graphs

The Seidel matrix S(G) of a graph G is the square matrix whose (i, j)-entry is equal to -1 or 1 if the i-th and j-th vertices of G are adjacent or non-adjacent, respectively, and is zero if i = j. The Seidel energy of G is the sum of the absolute values of the eigenvalues of S(G). We show that if G is regular of order n and of degree r ≥ 3, then for each k ≥ 2, the Seidel energy of the k-th iterated line graph of G depends solely on n and r. This result enables the construction of pairs of non-cospectral, Seidel equienergetic graphs of the same order.

Complete solution of equation [formula omitted] for the Wiener index of iterated line graphs of trees

Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. In Knor et al. (in press) we show that there is an infinite class T of trees T satisfying W(L3(T))=W(T), which disproves a conjecture of Dobrynin and Entringer. In this paper we prove that except the trees of T, there is no non-trivial tree T satisfying W(L3(T))=W(T). Consequently, for a tree T and i≥3, the equation W(Li(T))=W(T) holds if and only if T∈T and i=3.

Branch-bonds, two-factors in iterated line graphs and circuits in weighted graphs

A nontrivial path is called a branch if it has only internal vertices of degree two and end vertices of degree not two. A set S of branches of a graph G is called a branch cut if the deletion of all edges and internal vertices of branches of S results in more components than G. A minimal branch cut is called a branch-bond. In this paper, we found some relationships amongst branch-bonds, weighted graphs and two-factors in iterated line graphs. We also obtained some bounded number of components in two-factor of iterated line graphs by using the concept of branch-bonds.

Fundamental groups of iterated line graphs

The rank of the fundamental group, , of a connected graph is related to the Euler characteristic, , of by in this article, the Euler characteristic of the iterated line graph of and its complement is studied.

Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph

Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L k (G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ? k + 1 ? 4, then in G k , for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T, h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G, h(G) ≤ hc(G) ≤ max{h(G) + 1, k? + 1}, where k? is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.

Wiener index of iterated line graphs of trees homeomorphic to [formula omitted

This is fourth paper out of five in which we completely solve a problem of Dobrynin, Entringer and Gutman. Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. Moreover, denote by H a tree on six vertices, out of which two have degree 3 and four have degree 1. Let j≥3. In previous papers we proved that for every tree T, which is not homeomorphic to a path, claw K1,3 and H, it holds W(Lj(T))>W(T). Here we prove that W(L4(T))>W(T) for every tree T homeomorphic to H. As a consequence, we obtain that with the exception of paths and the claw K1,3, for every tree T it holds W(Li(T))>W(T) whenever i≥4.

Wiener index of iterated line graphs of trees homeomorphic to the claw K_1,3

Let G be a graph. Denote by L i ( G ) its i -iterated line graph and denote by W ( G ) its Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there exist a non-trivial tree T and i ≥ 3 such that W ( L i ( T )) = W ( T )? In a series of five papers we solve this problem. In a previous paper we proved that W ( L i ( T )) > W ( T ) for every tree T that is not homeomorphic to a path, claw K 1,3 and to the graph of "letter H ", where i ≥ 3. Here we prove that W ( L i ( T )) > W ( T ) for every tree T homeomorphic to the claw, T ≠ K 1,3 and i ≥ 4.

The Wiener index in iterated line graphs

For a graph G, denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. We prove that the function W(Li(G)) is convex in variable i. Moreover, this function is strictly convex if G is different from a path, a claw K1,3 and a cycle. As an application we prove that W(Li(T))≠W(T) for every i≥3 if T is a tree in which no leaf is adjacent to a vertex of degree 2, T≠K1 and T≠K2. This is the first in a series of papers which resolve the question, whether there exists a tree T for which W(Lk(T))=W(L(T)) for some k≥3, which was posed in [A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), 211–249].

The asymptotic behavior of some indices of iterated line graphs of regular graphs

In this paper, we consider the asymptotic behavior of the number of spanning trees and the Kirchhoff index of iterated line graphs and iterated para-line graphs (or clique-inserted graphs) of a regular graph G. We show that the asymptotic behavior of these indices (except the Kirchhoff index of the iterated para-line graphs) is independent of the structure of the regular graph G.

On a conjecture about Wiener index in iterated line graphs of trees

Let G be a graph. Denote by L i ( G ) its i -iterated line graph and denote by W ( G ) its Wiener index. Dobrynin and Melnikov conjectured that there exists no nontrivial tree T and i ≥ 3 , such that W ( L i ( T ) ) = W ( T ) . We prove this conjecture for trees which are not homeomorphic to the claw K 1 , 3 and H , where H is a tree consisting of 6 vertices, 2 of which have degree 3.

Upper Bounds for the Rainbow Connection Numbers of Line Graphs

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if for any two vertices of G there is a rainbow path connecting them. The rainbow connection number of G, denoted rc(G), is defined as the smallest number of colors such that G is rainbow connected. In this paper, we mainly study the rainbow connection number rc(L(G)) of the line graph L(G) of a graph G which contains triangles. We get two sharp upper bounds for rc(L(G)), in terms of the number of edge-disjoint triangles of G. We also give results on the iterated line graphs.

Outerplanarity of line graphs and iterated line graphs

The outerplanar index of a graph G is the smallest integer k such that the k th iterated line graph of G is non-outerplanar. In this note, we show: (i) the characterization of the forbidden subgraphs for graphs with outerplanar line graphs; (ii) that the outerplanar index of a graph is either infinite or at most 3. Furthermore, we give a full characterization of all graphs with respect to their outerplanar index.

Connectivity of iterated line graphs

Let k ≥ 0 be an integer and L k ( G ) be the k th iterated line graph of a graph G . Niepel and Knor proved that if G is a 4-connected graph, then κ ( L 2 ( G ) ) ≥ 4 δ ( G ) − 6 . We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ ( L 2 ( G ) ) ≥ 4 δ ( G ) − 6 . Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.

Closure, stability and iterated line graphs with a 2-factor

We consider 2-factors with a bounded number of components in the n -times iterated line graph L n ( G ) . We first give a characterization of graph G such that L n ( G ) has a 2-factor containing at most k components, based on the existence of a certain type of subgraph in G . This generalizes the main result of [L. Xiong, Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407–422]. We use this result to show that the minimum number of components of 2-factors in the iterated line graphs L n ( G ) is stable under the closure operation on a claw-free graph G . This extends results in [Z. Ryjáček, On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997) 217–224; Z. Ryjáček, A. Saito, R.H. Schelp, Closure, 2-factors and cycle coverings in claw-free graphs, J. Graph Theory 32 (1999) 109–117; L. Xiong, Z. Ryjáček, H.J. Broersma, On stability of the hamiltonian index under contractions and closures, J. Graph Theory 49 (2005) 104–115].

Distance spectra and distance energies of iterated line graphs of regular graphs

The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1) = ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G)? 2, and Fi,i = 1,2,3,4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n, diam(G)?2, and G does not have Fi,i=1,2,3,4, as an induced subgraph, then for k ?1, ED(Lk(G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.

Collapsible graphs and reductions of line graphs

A graph G is collapsible if for every even subset X ⊆ V ( G ) , G has a subgraph Γ such that G − E ( Γ ) is connected and the set of odd-degree vertices of Γ is X . A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G . In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. Lai, Reduced graph of diameter two, J. Graph Theory 14 (1) (1990) 77–87], and in [P.A. Catlin, Iqblunnisa, T.N. Janakiraman, N. Srinivasan, Hamilton cycles and closed trails in iterated line graphs, J. Graph Theory 14 (1990) 347–364].

Hamilton-connected indices of graphs

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131–148] defined h c ( G ) to be the least integer m such that the iterated line graph L m ( G ) is Hamilton-connected. Let diam ( G ) be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G . In this paper, we show that k − 1 ≤ h c ( G ) ≤ max { diam ( G ) , k − 1 } . We also show that κ 3 ( G ) ≤ h c ( G ) ≤ κ 3 ( G ) + 2 where κ 3 ( G ) is the least integer m such that L m ( G ) is 3-connected. Finally we prove that h c ( G ) ≤ | V ( G ) | − Δ ( G ) + 1 . These bounds are all sharp.

Formula omitted]-ordered hamiltonicity of iterated line graphs

A graph G of order n is k -ordered hamiltonian, 2 ≤ k ≤ n , if for every sequence v 1 , v 2 , … , v k of k distinct vertices of G , there exists a hamiltonian cycle that encounters v 1 , v 2 , … , v k in this order. In this paper, we generalize two well-known theorems of Chartrand on hamiltonicity of iterated line graphs to k -ordered hamiltonicity. We prove that if L n ( G ) is k -ordered hamiltonian and n is sufficiently large, then L n + 1 ( G ) is ( k + 1 ) -ordered hamiltonian. Furthermore, for any connected graph G , which is not a path, cycle, or the claw K 1 , 3 , there exists an integer N ′ such that L N ′ + ( k − 3 ) ( G ) is k -ordered hamiltonian for k ≥ 3 .

The existence of even factors in iterated line graphs

An even factor of a graph is a spanning subgraph of G in which all degrees are even, positive integers. In this paper, we characterize the claw-free graphs having even factors and then prove that the n -iterated line graph L n ( G ) of G has an even factor if and only if every end branch of G has length at most n and every odd branch-bond of G has a branch of length at most n + 1 .