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Acetalization of glycerol with aldehydes to form cyclic acetals is an industrially important reaction and is generally carried out using acid catalysts.

The theorem " the number of partitions of a positive integer n into distinct odd parts equals the number of partitions of n into parts = 2 and differing by 6 with strict inequality if a part is even" was first discovered and then proved by Krishnaswamy Alladi in the year 1999. The object of this paper is to give a different bijective proof of this result.

It is exciting to study and establish relationships between the physical properties and the molecular structure of chemicals and there is a scope for defining new topological indices. This paper aims to introduce a new topological index for graphs called Chelo index. The Chelo index of a graph G is the sum of five times order of G and two times the number of geodesics of length 3 minus the number of geodesics between peripheral vertices. We compute Chelo index for some standard graphs and observe the correlation between some physical properties and Chelo index for low alkanes. Also, we establish a formulae for computing the number of 176 graph geodesics in a graph and the Chelo index using the adjacency matrix

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