Geodesic
More Examples and Counterexamples for a Conjecture of Merrifield and Simmons
Publications de l'Institut Mathématique, _N_S_69 (2001) no. 83, p. 41 .

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Let $\sigma(G)$ be the number of independent-vertex sets of a graph $G$. Merrifield and Simmons conjectured that for any connected graph $G$ and any pair of its non-adjacent vertices $u$ and $v$, $\Delta_{uv}(G) := \sigma(G-u)\,\sigma(G-v) - \sigma(G)\,\sigma(G-u-v)$ is positive if the distance between $u$ and $v$ is odd, and negative otherwise. In earlier works by the authors the conjecture was shown to be true for trees, cycles and several other types of graphs, but a few counterexamples were discovered among dense graphs. We now prove that the conjecture is true for all bipartite and some non-bipartite connected unicyclic graphs, but not for all connected unicyclic graphs. Moreover, we find a graph $G$ for which $\Delta_{uv}(G)=0$.
Classification : 05C70 05C12 
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     author = {Yong Wang and Xueliang Li and Ivan Gutman},
     title = {More {Examples} and {Counterexamples} for a {Conjecture} of {Merrifield} and {Simmons}},
     journal = {Publications de l'Institut Math\'ematique},
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     publisher = {mathdoc},
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     year = {2001},
     zbl = {0997.05069},
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Yong Wang; Xueliang Li; Ivan Gutman. More Examples and Counterexamples for a Conjecture of Merrifield and Simmons. Publications de l'Institut Mathématique, _N_S_69 (2001) no. 83, p. 41 . http://geodesic.mathdoc.fr/item/PIM_2001_N_S_69_83_a6/