Abstract
Let T be an acyclic graph without perfect matching and Z ( T ) be its Hosoya index; let F n be the n th Fibonacci number. It is proved in this work that Z ( T ) ≤ 2 F 2 m F 2 m + 1 when T has order 4 m with the equality holding if and only if T = T 1 , 2 m − 1 , 2 m − 1 , and that Z ( T ) ≤ F 2 m + 2 2 + F 2 m F 2 m + 1 when T has order 4 m + 2 with the equality holding if and only if T = T 1 , 2 m + 1 , 2 m − 1 , where m is a positive integer and T 1 , s , t is a graph obtained by joining an isolated vertex with an edge to the ( s + 1 ) -th vertex (according to its natural ordering) of path P s + t + 1 .
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