On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index |
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Authors: | Jianping Ou |
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Institution: | (1) Department of Mathematics and Physics, Wuyi University, Jiangmen, 529020, China |
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Abstract: | Let G be an n-vertex unicyclic molecular graph and Z(G) be its Hosoya index, let F
n
be the nth Fibonacci number. It is proved in this paper that if G has girth l then Z(G) ≥ F
l+1+(n−l)F
l
+F
l-1, with the equality holding if and only if G is isomorphic to , the unicyclic graph obtained by pasting the unique non-1-valent vertex of the complete bipartite graph K
1,n-l
to a vertex of an l-vertex cycle C
l
. A direct consequence of this observation is that the minimum Hosoya index of n-vertex unicyclic graphs is 2n−2 and the unique extremal unicyclic graph is . The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined. |
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Keywords: | Hosoya index unicyclic molecular graph Fibonacci number matching |
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