Given a set of numbers, the k-SUM problem asks for a subset of k numbers that sums to zero. When the numbers are integers, the time and space complexity of k-SUM is generally studied in the word-RAM model; when the numbers are reals, the complexity is studied in the real-RAM model, and space is measured by the number of reals held in memory at any point. We present a time and space efficient de...

A sum graph G is a graph with a mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S. In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G. A grap...

Let D be a t-(v, k, λ) design and let Ni(D), for 1 ≤ i ≤ t, be the higher incidence matrix of D, a (0, 1)-matrix of size ( v i ) × b, where b is the number of blocks of D. A zero-sum flow of D is a nowhere-zero real vector in the null space of N1(D). A zero-sum k-flow of D is a zero-sum flow with values in {±1, . . . ,±(k− 1)}. In this paper we show that every non-symmetric design admits an int...

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers $$p=0,1,2$$ and $$|t|\le 1$$ . $$\begin{aligned} \sum _{k=1}^{\infty }\frac{H_{k-1}t^k}{k^p\left( {\begin{array}{c}n+k\\ k\end{array}}\right) } \quad \text{ }\quad }\frac{t^k}{k^p\left( }. \end{aligned}$$ also generalize relation between Stirling first kind Riemann zeta function to polyg...

As an analogous concept of a nowhere-zero flow for directed graphs, we consider zero-sum flows for undirected graphs in this article. For an undirected graph G, a zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the values of edges are less than k. We define the zer...

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 ...

This paper considers a degree sum condition sufficient to imply the existence of k vertexdisjoint cycles in a graph G. For an integer t ≥ 1, let σt (G) be the smallest sum of degrees of t independent vertices of G. We prove that if G has order at least 7k+1 and σ4(G) ≥ 8k−3, with k ≥ 2, then G contains k vertex-disjoint cycles. We also show that the degree sum condition on σ4(G) is sharp and co...