Score sets in oriented 3-partite graphs

Let D(U, V, W) be an oriented 3-partite graph with |U|=p, |V|=q and |W|= r. For any vertex x in D(U, V, W), let d x and d-x be the outdegree and indegree of x respectively. Define aui (or simply ai) = q r d ui - d-ui, bvj(or simply bj) = p r d vj - d-vj and Cwk (or simply ck) = p q d wk - d-wk as the scores of ui in U, vj in V and wk in Wrespectively. The set A of distinct scores of the vertices of D(U, V, W) is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai(2≤i≤n - 1) are even positive integers and an is any positive integer, then for n≥3, there exists an oriented 3-partite graph with the score set A = {a1,2∑i=1 ai,…,n∑i=1 ai}, except when A = {0,2,3}. Some more results for score sets in oriented 3-partite graphs are obtained.     

On Distance Laplacian Spectra of Certain Finite Groups

Acta Mathematica Sinica, English Series - For a finite group G, the power graph $$\cal{P}(G)$$ is a simple connected graph whose vertex set is the set of elements of G and two distinct vertices x...     

图中的度序列

给出了一个非减的非负整数序列是某个图的度序列的一个新刻划.     

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.     

Carbon-13 substituent chemical shifts of 4-substituted bicyclo[2.2.2]oct-1-yl cyanides

The carbon-13 substituent chemical shifts of 4-substituted bicyclo[2.2.2]oct-1-yl cyanides are a linear function of the substituent inductive parameters.     

On Potentially Graphical Sequences of G-E(H)

A loopless graph on n vertices in which vertices are connected at least by a and at most by b edges is called a(a,b,n)-graph. A(b,b,n)-graph is called(b,n)-graph and is denoted by K_n~b(it is a complete graph), its complement by K_n~b. A non increasing sequence π =(d_1,…,d_n) of nonnegative integers is said to be(a,b,n) graphic if it is realizable by an(a,b,n)-graph. We say a simple graphic sequence π=(d_1,…,d_n) is potentially K_4-K_2∪K_2-graphic if it has a a realization containing an K_4-K_2∪K_2 as a subgraph where K_4 is a complete graph on four vertices and K_2∪K_2 is a set of independent edges. In this paper, we find the smallest degree sum such that every n-term graphical sequence contains K_4-K_2∪K_2 as subgraph.     

Newton Method for Solving the Multi-Variable Fuzzy Optimization Problem

In this article, we propose the Newton method to find a non-dominated solution of an unconstrained multi-variable fuzzy optimization problem. For this purpose, we use the Hukuhara differentiability of fuzzy-valued functions and partial order relation on set of fuzzy numbers.     

Score sets ink-partite tournaments

The set S of distinct scores (outdegrees) of the vertices of ak-partite tournamentT(X ₁, X₂, ···, X_k) is called its score set. In this paper, we prove that every set of n non-negative integers, except {0} and {0, 1}, is a score set of some 3-partite tournament. We also prove that every set ofn non-negative integers is a score set of somek-partite tournament for everyn ≥k ≥ 2.