Let Xn and X*n be the finite sets {1,2,3,...,n} and {±1,±2,±3,..,±n} respectively. A map α: Xn→Xn is called a transformation on Xn We call α a signed transformation if α: Xn→X*n Let Tn and T˜n be the sets of full and signed full transformations on Xn respectively. The work, w(α) performed by a transformation α is defined as the sum of all the distances |i-iα| for each i ϵ dom(α) In this paper, we present a range for the values of w(α) for all α ϵ Tn. Further, we characterize elements of T˜n that attain minimum and maximum works and provide formulas for the values of these minimum and maximum.
Let \(\Gamma_{D_{2 n}}^{C}\) and \(E(\Gamma)\) denote the conjugate graph of a dihedral group of order \(2 n(n \in \aleph)\) and the energy of a graph respectively. The sum of the absolute values of the eigenvalues of an adjacency matrix's eigenvalues is the energy of a graph. In this paper, we use group representation of a dihedral group of order 2n with its conjugacy classes to explicitly design admissible conjugate graphs. We further introduced the general formula for the energy of conjugate graphs of dihedral groups in various circumstances. Also, we deduced the general formula for the conjugate graph of generalized dihedral groups of order 2n depending on the nature of n.
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