The Ordering Of Merrifield-Simmons Index And Hosoya Index For Graph Linked By Cycles

The Merrifield-Simmons index σ(G), of a graph G is defined as the total number of independent sets of the graph G and the Hosoya index μ(G),of a gra-ph G is defined as the total number of the matchings of the graph G. We denote the graph linked by cycles for a number of cycles with some special connections.In this paper, we construct graphs linked by cycles Zi(k) and Ti(k) (i=1,2,3, 4), which have specific connections. The special graph classes Zi(k) and Ti(k), are composed of four cycles with same order and five cycle sequences with different order, they are connected by four special ways "one vertex coincide"、 "one edge connect"、"one edge coincide"、"adjacent two vertices corresponding connect", and they have specific connections. Under some restrictions, we research graph linked by cycles Z; (k) and Ti(k) (i=1,2,3,4) about the ordering of Merrifield-Simmons index and Hosoya index, by using relevant properties of Fibonacci numbers and Lucas numbers.The results show that under some restrictions the graph classes Zi(k) and Ti(k) about the ordering of Merrifield-Simmons index and Hosoya index are on the contrary for each i∈{1,2,3,4), they are as follows:(1) The graph classes Zi(k) and Ti(k) about the ordering of Merrifield-Sim-mons indexMerrifield-Simmons index increases monotonically when k is odd, and its decreases monotonically when k is even. The maximum odd value’s index is less than the maximum even value’s index.(2) The graph classes Zi(k) and Ti(k) about the ordering of Hosoya indexHosoya index decreases monotonically when k is odd, and its index increases monotonically when k is even. The maximum odd value’s index is greater than the maximum even value’s index.