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Title

THE NUMBER OF SPANNING TREES OF POWER GRAPHS ASSOCIATED WITH SPECIFIC GROUPS AND SOME APPLICATIONS

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Abstract

 THE POWER GRAPH P (G) OF A GROUP G IS AN UNDIRECTED GRAPH WHOSE VERTEX SET IS G AND TWO VERTICES X; Y 2 G ARE ADJACENT IF AND ONLY IF⟨X⟩Í ⟨Y⟩ OR ⟨Y⟩ Í ⟨X⟩ (WHICH IS EQUIVALENT TO SAY X¹ Y AND XM=Y OR YM=X FOR SOME NON-NEGATIVE INTEGER M).CLEARLY, THE POWER GRAPH P (G) OF ANY GROUP G IS ALWAYS CONNECTED.THE NUMBER OF SPANNING TREES OF THE POWER GRAPH P (G) OF A GROUP G, WHICH IS DENOTED BY K(G) AND CALL THE TREE-NUMBER OF G, WILL BE INVESTIGATED FOR CERTAIN FINITE GROUPS G IN THIS TALK. INDEED, THE EXPLICIT FORMULA FOR THE TREE-NUMBER OF A CYCLIC GROUP OR A GENERALIZED QUATERNION GROUP IS OBTAINED. WE HAVE ALSO DETERMINED, UP TO ISOMORPHISM, THE STRUCTURE OF ANY FINITE GROUP G FOR WHICH K(G)<125.

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