For graphs G and H , a homomorphism from G to H is a function φ : V (G) → V (H), which maps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H . Many cases of graph homomorphism and locally injective graph homomorphism are NPcomplete, so there is little hope to design polynomial-time...

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph G as a mapping φ : V (G) → Z such that for any two adjacent vertices u and v the colour φ(u) is different from the colour σ(uv)φ(v), where is σ(uv) is the sign of the edge uv. The substantial part of Zaslavsky’s research concentrated on polynomial invariants related to signed graph colourings rather than on...

Let G be a simple graph of order n and let V (G) be its vertex set. Let c = a + b √ m and c = a − b√m, where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A = [cij ] is the conjugate adjacency matrix of the graph G if cij = c for any two adjacent vertices i and j, cij = c for any two nonadjacent vertices i and j, and cij = 0 if i =...

background: this study explored the relationship between socio-economic characteristics at the individual and neighbourhood levels, and wellbeing and lifestyle behaviours of young iranian women. methods: cluster convenience sampling was used to select 391 iranian women participated in this cross-sectional survey in shiraz, iran in 2013. a scale adapted from the british general household social ...

Let Ω = R 3 ∖ K ¯, where is an open bounded domain with smooth boundary Γ. V ( t ) e G b , ⩾ 0, be the semigroup related to Maxwell’s equations in dissipative condition ν ∧ E + γ x H > ∀ ∈ We study case when ≠ 1, Γ, and we establish a Weyl formula for counting function of eigenvalues polynomial neighbourhood negative real axis.

Abstract Single-machine scheduling where jobs have a penalty for being late or rejected altogether is an important (sub)problem in manufacturing, logistics, and satellite scheduling. It known to be NP-hard the strong sense, there no polynomial-time algorithm that can guarantee constant-factor approximation (unless P=NP). We provide exact fixed-parameter tractable slack maximum number of time wi...

A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their degrees are equal. In this paper, we investigate the set of all problems which are related to decomposition of graphs into regular, locally regular and/or loca...

An independent set in a graph G is a set of pairwise non-adjacent vertices, and the independence number α(G) is the cardinality of a maximum independent set. The independence polynomial of G is I(G; x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α(G), where sk equals the number of independent sets of size k in G (Gutman and Harary, 1983). If si = sα−i, 0 ≤ i ≤ α/2 , then I(G; x) is called palindromic. I...

In this paper, we study the Minimum Sum Coloring (MSC) problem on P4-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. First, we introduce the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph. N...

Our goal is to prove a generating theorem for the class E5 of all 5-regular simple planar graphs. This is a progress report. First, we will see the general information from Euler’s formula and the Discharge Method. Second, the basic graph operation D-operation will be introduced. Third, there are two cases to be discussed separately. A graph either contains an edge that is a part of three disti...