On The Extremal Values Of Some Monotonic Topological Indices In Graphs

Let G =(V,E)be a simple,undirected graph.A topological index is a function from the set of molecular graphs to the set of real number that is each real number represents a molecular graph.Various topological indices are proposed and researched by both theoretical chemists and mathematicians from different aspects.Let I(G)be a topological index of a graph G.If I(G + e)<I(G)(or I(G + e)>I(G),respectively)for each edge e(?)E(G),then I(G)is monotonically decreasing(or increasing,respectively)with addition of edges.In this thesis,based on the monotonically decreasing(or increasing,re-spectively)property of addition new edges,we consider the extremal problems with respect to some topological indices of connected graphs or bipartite graphs in terms of some graphic parameters.In Chapter 1,we introduce a brief background and some elementary ter-minologies,and some topological indices related to this thesis.In the first part of Chapter 2,we study the extremal values of monoton-ic indices among all connected bipartite graphs with given cut edges.In the second part,we consider the extremal values of monotonic indices among all connected bipartite graphs with given(edge)connectivity.Our results show that the corresponding extremal bipartite graphs of different topological in-dices are not completely coincide with each other.In Chapter 3,we determine the extremal values of some monotonic topo-logical indices in terms of the number of cut vertices,or the number of cut edges,or the vertex connectivity,or the edge connectivity of a graph,and characterize the corresponding extremal graphs among all graphs of order n.In Chapter 4,the extremal values of some monotonic topological indices among all connected graphs or bipartite graphs with given matching numbers axe obtained,the corresponding graphs are also determined,respectively.In Chapter 5,a unified approach to the extremal values of some monotonic topological indices among graphs in terms of bipartite vertex(edge)frustration index is established.Here,the bipartite vertex(edge)frustration index of graph G is defined as the minimum number of vertices(edges)whose deletion from G results in a bipartite graph.In the last Chapter,we summarize the main results in this work and give some prospects for further research.