On pages 338 and 339 in his first notebook (Notebooks (2 volumes), 1957), Ramanujan records eighteen values for a certain product of theta-functions. All these have been proved by B.C. Berndt, H.H. Chan and L.-C. Zhang (Proc. Edinburg Math. Soc. 40:583-612, 1997). The main purpose of this paper is to establish some new general theorems for the explicit evaluations of Ramanujan's product of theta-functions and some new explicit evaluations there from.

Recently Professor Chandrashekar Adiga et al [3] defined the minimum covering energy, E C (G) of a graph which depends on its particular minimum cover C. Motivated by this, we introduced minimum dominating energy of a graph E D (G) and computed minimum dominating energies of a star graph, complete graph, crown graph and cocktail graphs. Upper and lower bounds for E D (G) are established.

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Recently Prof.Chandrashekar Adiga et al. [1] have defined the minimum covering energy, E C (G) of a graph G which depends on its particular minimum cover C. Motivated by this paper, we introduced the concept of minimum covering distance energy E Cd (G) of a graph G and computed minimum covering distance energies of a star graph, complete graph, crown graph, bipartite graph and cocktail graphs. Upper and lower bounds for E Cd (G) are also established. Mathematics Subject Classification: 05C50, 05C69

In this paper we computed minimum covering Harary energies of a star graph, complete graph, crown graph, bipartite graph and cocktail party graphs. Upper and lower bounds for minimum covering Harary energies are also established.