Roger C. Entringer

Wiener Index of Trees: Theory and Applications

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.

On nonrepetitive sequences

Abstract We prove the following results: (1) There exists an infinite binary sequence having no identical adjacent blocks of length 3 or greater. (2) Every binary sequence of length greater than 18 has identical adjacent blocks of length 2 or greater. (3) Every infinite binary sequence has arbitrarily long adjacent blocks that are permutations of each other.

Largest induced subgraphs of the n-cube that contain no. 4-cycles

Every induced subgraph of the n-cube, Qn, with more than ⌈2n+13⌉ vertices is shown to contain a 4-cycle; this bound is sharp. The extremal graphs are characterized as translations of a specific subgraph of Qn.

Gossips and telegraphs

Suppose we have a group of n people, each possessing an item of information not known to any of the others and that during each unit of time each person can send all of the information he knows to at most other people. Further suppose that each of at most k other people can send all of the information they know to him. Determine the length of time required, f(n, k), so that all n people know all of the information. We show f(n, k) = ⌜logk+1n⌝. We define g(n, k) analogously except that no person may both send and receive information during a unit time period. We show ⌜logk+1n⌝≤g(n, k)≤2⌜logk+1n⌝ in general and further show that the upper bound can be significantly improvea in the cases k = 1 or 2. We conjecture g(n, k) = bk logk+1n+0(1) for a function bk we determine.

Average distance, minimum degree, and spanning trees

A graph G is perfectly orderable, if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph, if it contains no P4, 2K2, and C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev, and de Werra states that a graph is perfectly orderable, if it is the union of two threshold graphs. In this article, we investigate possible generalizations of the above theorem. Hoang has conjectured that, if G is the union of two graphs G1 and G2, then G is perfectly orderable whenever G1 and G2 are both P4-free and 2K2-free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to this conjecture, and we prove a special case of it: if G1 and G2 are two edge-disjoint graphs that are P4-free and 2K2-free, then the union of G1 and G2 is perfectly orderable.

An inequality for degree sequences

Abstract Let d 1 , d 2 , …, d n be the degree sequence of a simple graph and suppose p is a positive integer. We show that (∑ n i =1 d 1/ p i ) p ⩾ ∑ n i =1 d p i . Related ‘real’ inequalities, i.e., not graphdependent, are analyzed.

Arc Colorings of Digraphs

A pair of arcs of a digraph D are consecutive if the terminal point of one is the initial point of the other. An arc-coloring of D is an assignment of colors to the arcs so that no pair of consecutive arcs have the same color and the arc-chromatic number, c(D), is the minimum number of colors in an arc-coloring of D. It is shown that if Tn is the transitive tournament on n points then c(Tn) = {log2n} but [((n + 1)2] colors suffice if the color classes are required to be oriented trees. It is further shown that if D is the complete digraph (an arc from any point to any other point) on n points then c(D) ∼ log2n. Finally it is shown that if a digraph D is n-arc-colorable it is 2n(point) colorable and this bound is best.

Extremal values for ratios of distances in trees

The distance of a vertex u in a connected graph G is defined by σ(u) = ∑ v ϵ V(G)d(u, v) and the distance of G is given by σ(G) = 12 ∑ u ϵ V(G)σ(u). Extremal values for the ratios σ(T)σ(v), σ(T)σ(w), σ(w)σ(v), and σ(w)σ(u) are determined where T is a tree of order n, v is a centroid vertex of T, and w and u are end vertices of T.

Spanning cycles of nearly cubic graphs

Abstract The fact that a cubic hamiltonian graph must have at least three spanning cycles suggests the question of whether every hamiltonian graph in which each point has degree at least 3 must have at least three spanning cycles. We answer this in the negative by exhibiting graphs on n =2 m +1, m ≥5, points in which one point has degree 4, all others have degree 3, and only two spanning cycles exist.

Cycle-saturated graphs of minimum size

Abstract A graph G is called Ck-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in Ck-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c1n/k and n + c2n/k for some positive constants c1 and C2. Our results provide an asymptotic solution to a 15-year-old problem of Bollobas.