Chromaticity of Certain Bipartite Graphs

Since the introduction of the concepts of chromatically unique graphs and chromatically equivalent graphs, numerous families of such graphs have been obtained. The purpose of this thesis is to continue with the search of families of chromatically unique bipartite graphs. In Chapters 1 and 2, we define the concept of graph colouring, the associated chromatic polynomial and some properties of a chromatic polynomial. We also give some necessary conditions for graphs that are chromatically unique or chromatically equivalent. We end this chapter by stating some known results on the chromaticity of bipartite graphs, denoted as K(p,q)

An attempt to classify bipartite graphs by their chromatic Polynomial.

For the purpose of tackling the four-colour problem, Birkhoff (1912) introduced the chromatic polynomial of a map, denoted by P(M,A), which is a number of proper Acolouring of a map M. Whitney (1932), who established many fundamental results for it, later generalized the notion of a chromatic polynomial to that of an arbitrary graph. In 1968, Read asked whether it is possible to find a set of necessary and sufficient algebraic conditions for a polynomial to be the chromatic polynomial of some graph. In particular, Read asked for a necessary and sufficient condition for two graphs to be chromatically equivalent; that is, to have the same chromatic polynomial. In 1978, Chao and Whitehead defined a graph to be chromatically unique if no other graphs share its chromatic polynomial. Since then many researchers have been studying chromatic uniqueness and chromatic equivalence of graphs

Chromatically unique bipartite graphs with certain 3-independent partition numbers III

For integers p, q, s with p ≥ q ≥ 2 and s ≥ 0 , let ( ) 2 , K−s p q denote the set of 2_connected bipartite graphs which can be obtained from K(p,q) by deleting a set of s edges. In this paper, we prove that for any graph ( ) 2 G∈K−s p,q with p ≥ q ≥ 3 and 1 ≤ s ≤ q - 1 if the number of 3-independent partitions of G is 2p-1 + 2q-1 + s + 4, then G is chromatically unique. This result extends both a theorem by Dong et al.[2]; and results in [4] and [5]

Eccentric connectivity index of unicyclic graphs with application to cycloalkanes

Let G be a simple connected molecular graph. The eccentric connectivity index ξ(G) is defined as ξ (G) = ∑ν∈V(G)deg (ν)ec(ν), where deg(ν) denotes the degree of vertex v and ec(ν) is the largest distance between ν and any other vertex u of G. In this paper, we construct the general formulas for the eccentric connectivity index of unicyclic graphs with application to cycloalkanes

Eccentric connectivity index of some chemical trees

Let G = (V, E) be a simple connected molecular graph. In such a simple molecular graph, vertices represent atoms and edges represent chemical bonds, we denoted the sets of vertices and edges by V(G) and E(G), respectively. If d(u, v) be the notation of distance between vertices u, v ε V(G) and is defined as the length of a shortest path connecting them. Then, the eccentricity connectivity index of a molecular graph G is defined as ζ(G) = Σ vεv(G) deg(v)ec(v), where deg(v) is degree of a vertex v ε V(G), and is defined as the number of adjacent vertices with v. ec(v) is eccentricity of a vertex v ε V(G), and is defined as the length of a maximal path connecting to another vertex of v. In this paper, we establish the general formulas for the eccentricity connectivity index of some classes of chemical trees

The atom bond connectivity index of some trees and bicyclic graphs

The atom bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G)=∑uv∈E(G)√dv+du−2/dv⋅du, where d u denotes the degree of a vertex u in G. In this paper, we obtain the general formula for ABC index of some special, chemical trees, and bicyclic graphs

On the atom bond connectivity index of certain trees and unicyclic graphs

The atom-bond connectivity (ABC) index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G) = Σuvϵ √(du + dv-2) /du/dv, where du denotes the degree of a vertex u in G. In this paper, we obtain the general formula of the ABC index for certain trees and unicyclic graphs with their representing by Alkanes and Cycloalkanes