An injective map f : E(G) → {±1,±2, · · · ,±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f∗ : V (G) → Z − {0} defined by f∗(v) = ∑ e∈Ev f (e) is one-one, where Ev denotes the set of edges in G that are incident with a vetex v and f∗(V (G)) is either of the form { ±k1,±k2, · · · ,±k p 2 } or { ±k1,±k2, · · · ,±k p−1 2 } ∪ { ±k p+1 2 } according as ...

For a planar graph on n vertices we determine the maximum values for the following: 1) the sum of the m largest vertex degrees. 2) the number of vertices of degree at least k. 3) the sum of the degrees of vertices with degree at least k.

The Liouville function λ(n) is the completely multiplicative function whose value is −1 at each prime. We develop some algorithms for computing the sum T (n) = ∑n k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T (n) < 0. This answers a question originating in some work of Turán, who linked the behavior of T (n) to questions about the Riemann zeta function. ...

We call a sum a1 + a2 + · · ·+ ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ · · · ≤ ak and n = a1 + a2 + · · ·+ ak. For i = 1, 2, . . . , k, we call ai the i-th part of the sum a1 + a2 + · · ·+ ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1 + a2 + · · · + ak and b1 + b2 + · · ·+ bk strongly inter...

Let there be given n ordinals al, a2, • • • , an . It is well known that every ordinal can be written uniquely as the sum of indecomposable ordinals. (An ordinal is said to be indecomposable if it is not the sum of two smaller ordinals .) Denote by ¢(a) the largest of these indecomposable ordinals belonging to a . (4(a) may have a coefficient c in the decomposition of a.) Put y=minis . 0(ai), a...

A ranked poset is called strongly Sperner if the size of k-family cannot exceed the sum of k-largest Whitney numbers. In the sense of a function ordering, a function f is (weakly) majorized by g if the the sum of k-largest values in f cannot exceed the sum of k-largest values in g. Two definitions arise from different contexts, but they have strong similarities in their own meanings. Furthermor...

We find all nonnegative integer r, s, p for which the sum ∑sn k=rn (pn k ) has closed form.

Supplementary materials to the main paper. 1. Proofs Proof of Proposition 4 Proof. We have E[Sk] = ∑ i∈V Ezi [1{zi=k∧yi=k}] = ∑ i∈V pi(k)1{yi=k}. Likewise we have E[Tk] = ∑ i∈V Ezi [1{zi=k∨yi=k}] = ∑ i∈V Ezi [1{zi=k} + 1{yi=k} − 1{zi=k∧yi=k}] = ∑ i∈V (pi(k) + 1{yi=k} − pi(k) 1{yi=k}) = ∑ i∈V (1{yi=k} + pi(k) 1{yi 6=k}). Proof of Proposition 5 Proof. In (5) the first central moment 〈Tk〉 is alway...

Finding the sub-vector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k sub-vectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n+k) time algorithm for the k maximal sums problem. We use this algorithm to obtain algorithms solving the two-dimensional k max...