Yeong-Nan Yeh

The Wiener Polynomial of a Graph

rn The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincar6 polynomial of a finite Coxeter group. 0 1996 John Wiley & Sons, Inc.

On the sum of all distances in composite graphs

Let G, x G,, G, + G,, G, [G2 J, G, 0 G, and G, {G,} be the product, join, composition, corona and cluster, respectively, of the graphs G, and G,. We compute the sum of distances between all pairs of vertices in these composite graphs.

Graph homotopy and Graham homotopy

Abstract Simple-homotopy for cell complexes is a special type of topological homotopy constructed by elementary collapses and elementary expansions. In this paper, we introduce graph homotopy for graphs and Graham homotopy for hypergraphs and study the relation between the two homotopies and the simple-homotopy for cell complexes. The graph homotopy is useful to describe topological properties of discretized geometric figures, while the Graham homotopy is essential to characterize acyclic hypergraphs and acyclic relational database schemes.

Generalized Dyck paths

Abstract It is well known (see [3, 6, 9, 10, 11]) that Dyck paths are in bijection with “Dyck words”, “ballot sequences”, “well formed sequences of parentheses”, “2-lines standard-tableaux”, “binary trees”, “ordered trees”; all these are counted by Catalan numbers. In the present text, we replace the north-east steps of a Dyck path by steps from an arbitrary finite multi-set l of vectors with integral coordinates in the plane. In order to study these generalized Dyck paths, called A-paths, we have to introduce many closely related families of paths. The corresponding (multi-variable) generating functions satisfy an intricate system of algebraic equations which leads to a polynomial equation satisfied by A. For example when l is {(1, 1)} (respectively {(1,2), (2,1); {(1,3), (3,1)}) this polynomial equation is of degree 2 (resp. 4, 8). More generally when l ={u 1 , u 2 ,…, u m } where uj = (rj, j) then A = A(u1, u2,…, um) satisfies a polynomial equation of degree 2m with coefficients in Z [u1, u2,…, um].

The behavior of Wiener indices and polynomials of graphs under five graph decorations

Abstract The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q -analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial , extends this concept by trying to capture the complete distribution of distances in the graph. Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.

Polynomials with real zeros and Pólya frequency sequences

Let f(x) and g(x) be two real polynomials whose leading coefficients have the same sign. Suppose that f(x) and g(x) have only real zeros and that g interfaces f or g alternates left of f. We show that if ad ≥ bc then the polynomial (bx + a)f(x) + (dx + c)g(x) has only real zeros. Applications are related to certain results of Brenti (Mem. Amer. Math. Soc. 413 (1989)) and transformations of Polya-frequency (PF) sequences. More specifically, suppose that A(n, k) are nonnegative numbers which satisfy the recurrence A(n,k) = (rn + sk + t)A(n - 1,k - 1) + (an + bk + c)A(n - 1,k) for n ≥ 1 and 0 ≤ k ≤ n, where A(n,k) = 0 unless 0≤k≤n. We show that if rb≥as and (r+s+t)b≥(a+c)s, then for each n≥0, A(n, 0), A(n, 1),..., A(n, n) is a PF sequence. This gives a unified proof of the PF property of many well-known sequences including the binomial coefficients, the Stirling numbers of two kinds and the Eulerian numbers.

Catalan and Motzkin numbers modulo 4 and 8

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.

Enumeration of subtrees of trees

Let T be a weighted tree. The weight of a subtree T1 of T is defined as the product of weights of vertices and edges of T1. We obtain a linear-time algorithm to count the sum of weights of subtrees of T . As applications, we characterize the tree with the diameter at least d, which has the maximum number of subtrees, and we characterize the tree with the maximum degree at least ∆, which has the minimum number of subtrees.

Taylor expansions for Catalan and Motzkin numbers

In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae associated with Catalan and Motzkin numbers. In particular, our expansion for Catalan number provides a simple proof of the classic Chung?Feller theorem; similar result for the Motzkin paths with flaws is also given.

Refined Chung-Feller theorems for lattice paths

In this paper we prove a strengthening of the classical Chung-Feller theorem and a weighted version for Schroder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating schroder paths with flaws.