Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), ..., d(v, Sk)), where d(v, Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if the k-vectors r(v|Π), v ∈ V (G), are distinct. The minimum k for which there is a re...

It was conjectured by Hajós that graphs containing no K 5 -subdivision are 4-colorable. Previous results show any possible minimum counterexample to Hajós' conjecture, called graph, is 4-connected but not 5-connected. In this paper, we if a graph admits 4-cut or 5-cut with planar side then the must be small contains special wheel. This step in our effort reduce conjecture Four Color Theorem.

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

Inspired by the chemical applications of higher-order connectivity index (or Randic index), we consider here the higher-order first Zagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order first Zagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order first Zag...

K e y w o r d s R a m s e y number, Tree, Wheel. 1. I N T R O D U C T I O N All graphs considered in this paper are finite simple graphs without loops. For two given graphs G1 and G2, the Ramsey number R(G1, G2) is the smallest positive integer n such that for any graph G of order n, either G contains G1 or G contains G2, where G is the complement of G. Let G be a graph. We use V(G) and E(G) to...

This paper first considers several types of additive bases. A typical problem is to find nv(k), the largest n for which there exists a set {0 al < a2 <" < ak} Of distinct integers modulo n such that each in the range 0 =<-< n can be written at least once as mai + aj (modulo n) with </'. For example, nv(8) 24, The other problems arise if at least is changed to at most, or </' to-</', or if the w...

Corresponding to the game chromatic number of graphs, we consider in this paper the game chromatic index χg of graphs, which is defined similarly, except that edges, instead of vertices of graphs are colored. Upper bounds for trees and wheels are given.

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The greatest eigenvalue λ1 of D(G) is called the distance spectral radius of the graph G and is denoted by λ1(G). A simple connected graph G is called a 2-partite distance regular graph if there exists a partition V1 ∪V2 of the vertex set of G such that for i = 1, 2 and an...