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Recently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows:

∏ 1 = ∏ 1 ( G ) = ∏ v ∈ V ( G ) d G ( v ) 2 , ∏ 2 = ∏ 2 ( G ) = ∏ u v ∈ E ( G ) d G ( u ) d G ( v ) .

These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs.

MSC:05C05, 05C90, 05C07.

In this paper, we prove the existence and uniqueness of a fixed point for certainα-admissible contraction mappings. Our results generalize and extend some well-known results on the topic in the literature. We consider some examples to illustrate the usability of our results.

MSC:46N40, 47H10, 54H25, 46T99.