The Hosoya Index Of A Kind Of Graphs With M-matching

Hosoya index was put forward and conducted research by Japanese chemistHosoya in1971. It was defined as the total number of matchings of the graph, whichcontained the empty set, and denoted by Z(G). Hosoya index are closely linked withthe boiling point of the material, entropy, calculation of chemical bonds and chemicalstructures, etc..Ever since the Hosoya index was introduced, there has already obtainedabundant research findings. Gutman proved that linear hexagonal chain is the onlychain that has the smallest Hosoya index among all hexagonal chain; Zhang and Tianhas found the extremal graph with the second minimal Hosoya index among allhexagonal chain; in addition, there are many academics studied the characteristics ofdouble hexagonal chain and hexagonal spider chains. Li has been gained the extremalgraphs with the smallest and the second smallest Hosoya indices in the catacondensedsystem. For the trees of n vertices, Dankelmann determined the road of n vertices isthe tree with the largest Hosoya index, and the star of n vertices is the tree with thesmallest Hosoya index; Gutman obtained the extremal graphs with the largest to thefourth-largest Hosoya indices. For the trees with the given matching, Hou presentedthe extremal graphs with the smallest and the second smallest Hosoya indices. For thetrees of n vertices, Li and Zhou got the extremal graphs with the third-largest, and thefifth-largest to the seventh-largest Hosoya indices, respectively. For the unicyclicgraphs,Li has found the extremal graphs with the smallest to the Ninth smallestHosoya indices. For the unicyclic graphs with the given matching,Yu and Tiancharacterized the extremal graphs with the smallest and the second smallest Hosoyaindices;Ye characterized the unicyclic graphs with the third smallest to sixth smallestHosoya indices. For the bicyclic graphs, Deng got the bicyclic graph with the smallestHosoya index. For the bicyclic graphs with the given matching, Yu and Tiancharacterized the extremal graph with the smallest Hosoya index. For the tricyclegraphs,Li and Zhu studied the smallest Hosoya index and obtained the extreme valuesand their corresponding extremal graphs.In this paper, on the basis of existing results, first of all we characterized thesecond smallest and the third smallest Hosoya indices among the bicyclic graphs with the given matching, and from this we got the second smallest and the third smallestHosoya indices among the vertex-gluing bicyclic graphs with the given matching,finally we characterized the second smallest and the third smallest Hosoya indicesamong the connected graphs with the given matching and the given cyclomaticnumber.