Abstract
Let G be a unicyclic n -vertex graph and Z ( G ) be its Hosoya index, let F n stand for the n th Fibonacci number. It is proved in this paper that Z ( G ) ≤ F n + 1 + F n − 1 with the equality holding if and only if G is isomorphic to C n , the n -vertex cycle, and that if G ≠ C n then Z ( G ) ≤ F n + 1 + 2 F n − 3 with the equality holding if and only if G = Q n or D n , where graph Q n is obtained by pasting one endpoint of a 3-vertex path to a vertex of C n − 2 and D n is obtained by pasting one endpoint of an ( n − 3 ) -vertex path to a vertex of C 4 .
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