This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees. Let G = (V,E) be a connected graph on n vertices, V = {1, 2, . . . , n}, with adjacency matrix A. For any integer k ≥ 1, let a ij denote ...

We show that the random number Tn of triangles in a random graph on n vertices, with a strict constraint on the total number of edges, admits an expansion Tn = an 3 + bn + Fn, where a and b are numbers, with the mean 〈Fn〉 = O(n) and the standard deviation σ(Tn) = σ(Fn) = O(n ). The presence of a ‘surface term’ bn has a significance analogous to the macroscopic surface effects of materials, and ...

We study the influence of a possible coherent component in the boson source on the two-, threeand n-particle correlation functions in a generalized core/halo type of boson-emitting source. In particular, a simple formula is presented for the strengh of the n-particle correlation functions for such systems. Graph rules are obtained to evaluate the correlation functions of arbitrary high order. T...

Paul Seymour conjectured that any graph G of order n and minimum degree at least k k+1n contains the k th power of a Hamilton cycle. We prove the following approximate version. For any > 0 and positive integer k, there is an n0 such that, if G has order n ≥ n0 and minimum degree at least ( k k+1 + )n, then G contains the kth power of a Hamilton cycle. c © 1998 John Wiley & Sons, Inc. J Graph Th...

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

The Waring number of the integers modulo m with respect to k-th powers, denoted by ρ(m, k), is the smallest r such that every integer is a sum of r k-th powers modulo m. This number is also the diameter of an associated Cayley graph, called the Waring graph. In this paper this number is computed when m is a power of 2. More precisely the following result is obtained: Let n, s and b be natural n...

An explicit description is giv e n for th e uniqu e gra ph with as few arcs (eac h bearin g a positive length) as pos s ibl e, whi c h has a presc rib ed mat rix of s hortest-p ath di stan ces be twee n pa irs of distinct vertices. The sam e is d one in th e case wh e n the ith diago na l matrix e ntr y, in s te ad o f be ing zero , represents th e. le ngth of a s hort est c losed path co ntain...