Unimodality And Genus Distributions Of Certain Types Of Graphs

In the paper,genus distributions of a general type of Terraces and the unimodality of some ladders are studied.Through this paper, a surface、a graph and an embedding are always regarded as an orientable cycle of letters、a connected graph and an orientable2-cell embed-ding. Since the genus distribution of a graph was introduced in1987,it gives a rise to many scholars’ attention. First,genus distributions of some simple types of graphs were calculated,such as:closed-end ladders、Mobius ladders、Ringel ladders、cobble stone paths.Then, the graphs gets more sophisticated, which are general ladders、3-regular graphs and4-regular graphs,etc. Generally, methods applied are combinatorial ways、Jackson’s formula、matrices、surface generating and sorting surface based on joint trees and partial genus distributions.We calculated genus distributions of a new type of sets of surfaces. At the basis of joint trees introduced by Yanpei Liu, by using the method which sorts the embedding surfaces of these graphs, the genus distribution of the orientable embeddings for a new general type of graphs are provided by applying those of sets of surfaces. Genus distributions of the type of graphs P3□Pn are deduced as a special case, whose genus distributions were provided by Gross etc in2012. Besides, we get genus distributions of some ladders.At last, We give the result of the unimodality of genus distribution of certain lad-ders.In chapter1, we recall related concepts and conclusions of embeddings and genus distributions for graphs.In chapter2, first of all, we get recursion expressions of genus distributions of certain sets of surfaces.By sorting the embedding surfaces, genus distributions of a general type of Terraces are a linear combination of those for sets of surfaces. Then, we get the recursion expressions of genus distributions for these graphs and program to solve them.In chapter3, the paper mainly concerns unimodality and log-concavity of poly-nomial sequences there exist dependent recurrence relations among them.In Section2, several criteria are presented to determine unimodality of a sequence of numbers gen- erated from finite unimodal sequences of numbers. In Section3, genus distributions for sets of ladder surfaces are recalled to be unimodal or even log-concave. Then uni-modality of genus distributions for some ladders can be verified. In the section4, the peaks of of genus distributions for some ladders are determined.