The Harary index of ordinary and generalized quasi-tree graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v∈V(G) such that G?v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k-generalized quasi-tree graph to be a connected graph G with a subset V _k ?V(G) where |V _k |=k such that G?V _k is a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k=2.     

A local characterization of bounded clique-width for line graphs

It is shown that a line graph G has clique-width at most 8k+4 and NLC-width at most 4k+3, if G contains a vertex whose non-neighbours induce a subgraph of clique-width k or NLC-width k in G, respectively. This relation implies that co-gem-free line graphs have clique-width at most 14 and NLC-width at most 7.It is also shown that in a line graph the neighbours of a vertex induce a subgraph of clique-width at most 4 and NLC-width at most 2.     

Queue layouts of iterated line directed graphs

In this paper, we study queue layouts of iterated line directed graphs. A k-queue layout of a directed graph consists of a linear ordering of the vertices and an assignment of each arc to exactly one of the k queues so that any two arcs assigned to the same queue do not nest. The queuenumber of a directed graph is the minimum number of queues required for a queue layout of the directed graph.We present upper and lower bounds on the queuenumber of an iterated line directed graph L^k(G) of a directed graph G. Our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the results on the queuenumber of L^k(G), it is shown that for any fixed directed graph G, L^k(G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L^k(G). These results are also applied to specific families of iterated line directed graphs such as de Bruijn, Kautz, butterfly, and wrapped butterfly directed graphs. In particular, the queuenumber of k-ary butterfly directed graphs is determined if k is odd.     

The generating function of Whitworth runs

Let G be a graph. The number of ways of selecting k vertices in G such that the subgraph induced by the k selected vertices containing l edges may be considered as Whitworth runs. For two arbitrary graphs G₁ and G₂ we show that the generating function of G₁ can be written as a sum of the generating function of G₂. As an application we derive a difference equation satisfied by the generating function of a line graph and that of a cycle graph. Two independent solutions in the closed-form are found. One is equivalent to the Whitworth bracelet problem with two colors. Furthermore, a line and a cycle graph and a two-line graph have been studied. We show that all solutions can be written as a sum of the solutions for single-line cases.     

On stability of the hamiltonian index under contractions and closures

The hamiltonian index of a graph G is the smallest integer k such that the k‐th iterated line graph of G is hamiltonian. We first show that, with one exceptional case, adding an edge to a graph cannot increase its hamiltonian index. We use this result to prove that neither the contraction of an A_G(F)‐contractible subgraph F of a graph G nor the closure operation performed on G (if G is claw‐free) affects the value of the hamiltonian index of a graph G. AMS Subject Classification (2000): 05C45, 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

Sandpile groups and spanning trees of directed line graphs

We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.     

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.     

On n-Hamiltonian line graphs

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph L^m(G) of G is defined to be L(L^m-1(G)). A graph G of order p ≥ 3 is n-Hamiltonian, 0 ≤ n ≤ p ? 3, if the removal of any k vertices, 0 ≤ k ≤ n, results in a Hamiltonian graph. It is shown that if G is a connected graph with δ(G) ≥ 3, where δ(G) denotes the minimum degree of G, then L²(G) is (δ(G) ? 3)-Hamiltonian. Furthermore, if G is 2-connected and δ(G) ≥ 4, then L²(G) is (2δ(G) ? 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K(1, 3) and for any positive integer n, the existence of an integer k such that L^m(G) is n-Hamiltonian for every m ≥ k is exhibited. Then, for the special case n = 1, bounds on (and, in some cases, the exact value of) the smallest such integer k are determined for various classes of graphs.     

Star complements and exceptional graphs

Let G be a finite graph of order n with an eigenvalue μ of multiplicity k. (Thus the μ-eigenspace of a (0,1)-adjacency matrix of G has dimension k.) A star complement for μ in G is an induced subgraph G-X of G such that |X|=k and G-X does not have μ as an eigenvalue. An exceptional graph is a connected graph, other than a generalized line graph, whose eigenvalues lie in [-2,∞). We establish some properties of star complements, and of eigenvectors, of exceptional graphs with least eigenvalue −2.     

Connectivity of iterated line graphs

Let k≥0 be an integer and L^k(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L²(G))≥4δ(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(L²(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.     

A note on cycle spectra of line graphs

We show that line graphs G=L(H) with σ₂(G)≥7 contain cycles of all lengths k, 2rad(H)+1≤k≤c(G). This implies that every line graph of such a graph with 2rad(H)≥Δ(H) is subpancyclic, improving a recent result of Xiong and Li. The bound on σ₂(G) is best possible.     

Iteratedk-line graphs

For integersk≥2, thek-line graph L_k(G) of a graph G is defined as a graph whose vertices correspond to the complete subgraphs onk vertices in G with two distinct vertices adjacent if the corresponding complete subgraphs have 1 common vertices inG. We define iteratedk-line graphs byL _k ⁿ (G) ?L _k (L _k ^n?1 (G), whereL _k ⁰ (G) ?G. In this paper the iterated behavior of thek-line graph operator is investigated. It turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and k≥4.     

Edge-connectivity and edge-superconnectivity in sequence graphs

Given an integer k?1 and any graph G, the sequence graph S_k(G) is the graph whose set of vertices is the set of all walks of length k in G. Moreover, two vertices of S_k(G) are joined by an edge if and only if their corresponding walks are adjacent in G.In this paper we prove sufficient conditions for a sequence graph S_k(G) to be maximally edge-connected and edge-superconnected depending on the parity of k and on the vertex-connectivity of the original graph G.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Acyclic chromatic indices of planar graphs with large girth

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree Δ has a^′(G)=Δ if there exists a pair (k,m)∈{(3,11),(4,8),(5,7),(8,6)} such that G satisfies Δ≥k and g(G)≥m.     

On maximum planar induced subgraphs

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G^′. In this case G^′ is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G)?k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G)?k. We prove that unless P=NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.     

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω_c(G-x)>ω(G-x) for some x∈V(G), then G=K_(2k+1)/2+x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.     

A note on the strong edge-coloring of outerplanar graphs with maximum degree 3

A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, are assigned different colors. The strong chromatic index of G is the smallest integer k for which G has a strong k-edge-coloring. In this paper, we have shown that the strong chromatic index is no larger than 6 for outerplanar graphs with maximum degree 3.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.