An injective map from the vertex set of a graph G-its order may not be finite-to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is called arithmetic (geometric) if it admits an arithmetic (a geometric) labeling. In this article, we show...

For k = 1,2,... let $H_k$ denote the harmonic number ${\sum }_{j=1}^{k}1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ${\sum }_{k=1}^{p-1}\left(H_k\right)/\left(k{2}^k\right)\equiv 7/24p{B}_{p-3}\left(modp²\right)$, ${\sum }_{k=1}^{p-1}\left(H_{k,2}\right)/\left(k{2}^k\right)\equiv -3/8B_{p-3}\left(modp\right)$, and ${\sum }_{k=1}^{p-1}\left(H{²}_{k,2n}\right)/\left(k^{2n}\right)\equiv (\left(\genfrac{}{}{0pt}{}{6n+1}{2n-1}\right)+n)/(6n+1)p{B}_{p-1-6n}\left(modp²\right)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}:={\sum }_{j=1}^{k}1/\left(j^m\right)$.

Families of all sets of independent vertices in graphs are investigated. The problem how to characterize those infinite graphs which have arithmetically maximal independent sets is posed. A positive answer is given to the following classes of infinite graphs: bipartite graphs, line graphs and graphs having locally infinite clique-cover of vertices. Some counter examples are presented.

A composition of a positive integer n is a finite sequence π1π2...πm of positive integers such that π1+...+πm = n. Let d be a fixed number. We say that we have an ascent of size d or more (respectively, less than d) if πi+1 ≥ πi+d (respectively, πi < πi+1 < πi + d). Recently, Brennan and Knopfmacher determined the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. In this paper, we find an explicit formula for the multi-variable...