The subtree number index STN(G) of a simple graph G is the number of nonempty subtrees of G. It is a structural and counting topological index that has received more and more attention in recent years. In this paper we first obtain exact formulas for the expected values of subtree number index of random polyphenylene and spiro chains, which are molecular graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover, we establish a relation between the expected values of the subtree number indices of a random polyphenylene and its corresponding hexagonal squeeze. We also present the average values for subtree number indices with respect to the set of all polyphenylene and spiro chains with n hexagons.

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A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G = C m ∨ P n ( m ≥ 3 and n ≥ 1 ) be the graph join of C m (the cycle with m vertices) and P n (the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable.

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