| In this paper, based on the previous study of the index in chemical graphs we determinedthe extremal graphs of linear hexagonal spider about Merrifield - Simmons index andsummarize the relation of Merrifield-Simmons index and Hosoya index in general graphs.We also do some research on a famous conjecture about Randic′index and average distance.The thesis is arranged as follows:Section One is devoted to the introduction of some basic concepts and symbols in chemicalgraph theory, simply reviewed the development history , current studies and some problemswaiting for solving in the field of hexagonal spider of Merrifield - Simmons index andsome related result of Randic′index in chemical graphs.In Section Two, we study the extremal graph of linear hexagonal spider, and obtain thefollowing result: if the sum of the legs of the linear hexagonal spiders are the same value,the graphs class L(1,1,n1 +n2 +n3 -2) attains the maximal Merrifield-Simmons index.In Section Three, we study the conjecture of Randic′index and average distance: For allconnected graphs G of order n,and the bound is sharp for all n≥1. As these two parameters are related to n of therelationship between the relevant variables associated with no fixed formula and the rangeof under uniform conditions. So we mainly study the special cases when G are some graphclasses such as Pn, regular graphs,Θgraph, for these graphs the conjecture is true andwe can obtain the following results: among all the connected graphs G of order n≥2, ifR(G)≥R(Pn), then the conjecture is true for G. Moreover this result is only a su-cientcondition, the graph class of unicyclic graphs of order n≥4 is the example.
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