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Abstract: The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let be the set of connected -vertex graphs with connectivity at most . In this note, we characterize the extremal (maximal and minimal) graphs from with respect to the Hosoya index and the Merrifield–Simmons index, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we present some sharp upper bounds for the number of spanning trees of a connected graph in terms of its structural parameters such as the number of vertices, the number of edges, maximum vertex degree, minimum vertex degree, connectivity and chromatic number. [Copyright &y& Elsevier]