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 ...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

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...

For a century ago, one of the most famous problems in mathematics was to prove the FourColor Problem. During the period that the Four-Color Problem was unsolved, which spanned more than a century, many approaches were introduced with the hopes that they would lead to a solution of this famous problem. In 1913, Birkhoff 1 defined a function P M,λ that gives the number of proper λ-colorings of a ...

We measured the responses of 305 neurons in striate cortex to moving sinusoidal gratings modulated in chromaticity and luminance about a fixed white point. Stimuli were represented in a 3-dimensional color space defined by 2 chromatic axes and a third along which luminance varied. With rare exceptions the chromatic properties of cortical neurons were well described by a linear model in which th...

Let G be a graph whose each component has order at least 3. Let s : E(G) → Zk for some integer k ≥ 2 be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring c : V (G) → Zk defined by c(v) = ∑ e∈Ev s(e) in Zk, (where the indicated sum is computed in Zk and Ev denotes the set of all edges incident to v) results in a proper vertex col...

The Minimum Sum Coloring Problem (MSCP ) is derived from the Graph Coloring Problem (GCP ) by associating a weight to each color. The aim ofMSCP is to find a coloring solution of a graph such that the sum of color weights is minimum.MSCP has important applications in fields such as scheduling and VLSI design. We propose in this paper new upper bounds of the chromatic strength, i.e. the minimum ...

A graph G is called a sum graph if there is a so-called sum labeling of G, i.e. an injective function l : V (G) → N such that for every u, v ∈ V (G) it holds that uv ∈ E(G) if and only if there exists a vertex w ∈ V (G) such that l(u) + l(v) = l(w). We say that sum labeling l is minimal if there is a vertex u ∈ V (G) such that l(u) = 1. In this paper, we show that if we relax the conditions (ei...

We prove some Schur positivity results for the chromatic symmetric function X G of a (hyper)graph G, using connections to group algebra group. The first such connection works (hyper)forests F: we describe coefficients F in terms eigenvalues product Hermitian idempotents algebra, one factor each edge (a more general formula similar shape holds all chordal graphs). Our main application this techn...