Abstract
Let m(G,k) denote the number of matchings of cardinality k in a graph G. A quasi-order ⪯ is defined by writing G⪯H whenever m(G,k)≤m(H,k) holds for all k. We consider the set G1(n,γ) of connected graphs with n vertices and γ cut vertices as well as the set G2(n,γ) of connected graphs with n vertices and γ cut edges. We determine the greatest and least elements with respect to this quasi-order in G1(n,γ) and the greatest element in G2(n,γ) for all values of n and γ. As corollaries, we find that these graphs maximize (resp. minimize) the Hosoya index and the matching energy within the respective sets.
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