The Uniqueness Of Chromaticity Of Some Kinds Of Graphs And Maximal Energy Of Trees

In 1978, Chao and Whitehead defined a graph to be chromatically unique if no other graphs shar its chromatic polynomial. It is well-know that the chromatic polynomial of a graph is one of basic tools for studying the chromaticity of graphs. Since then various families results of this region have been found.For a graph G with p vertices, if G₀ is a spanning subgraph of G and each component of G₀ is a complete graph. Then G₀ is called an ideal subgraph of G. Let b_i(G) denote the number of ideal subgraph G₀ of G with p - i components. It is clear that b₀(G) = 1,b₁(G) = q(G). In 1987, Liu introduced the adjoint polynomial of G as follows: h(G,x) =∑_k=0^p-1b_i(G)x^p-i . For convenience, we simply denote h(G,x) by h(G). Two graphs G and H are said to be adjoint equivalent, denoted by G ^h H, if h(G,x) = h(H,x). The notion of the adjoint polynomial succeeded in the study of the chromaticity of graphs.The Hosoya index is an important topological parameter to study the relation between molecular structure and physical and chemical properties of certain hydrocarbon compound, such as the correlation with boiling points,entropies, calculated bond orders. Now there is its another application to solve the chromaticity of graphs.In chemistry, the energy of a given molecular graph is of interest since it is closely related to the totalÏ€-electron energy of the molecular represented by that graph in [21,22]. We know the introduce of the adjoint polynomial succeeded in study of the chromaticity of graphs, in fact, some properties of adjoint polynomial are useful in comparing with the energies of different graphs.In this thesis, we investigate the chromaticity of some kinds of graphs by some properties of adjoint polynomial and Hosoya index and study maximal energy of one kind of trees by adjoint polynomial.