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This paper focuses on the power graph of dihedral groups of order 2n, D2n, where n ≥ 3. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices. [ABSTRACT FROM AUTHOR]

This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n, where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness matrix. The result is that the obtained energy is always twice its spectral radius and is never an odd integer. Moreover, it is classified as hypoenergetic. [ABSTRACT FROM AUTHOR]

Let ΓG and ΓG be the commuting and non-commuting graphs on a finite group G, respectively, having G\Z(G) as the vertex set, where Z(G) is the center of G. The order of ΓG and ΓG is ΙG\Z(G)Ι, denoted by m. For ΓG, the edge joining two distinct vertices vp, vq ∈ G\Z(G) if and only if vpvq ≠= vqvp, on the other hand, whenever they commute in G, vp and vq are adjacent in ΓG. The degree subtraction matrix (DSt) of ΓG is denoted by DSt(ΓG), so that its (p, q)-entry is equal to dvp - dvq, if vp ≠ = vq, and zero if vp = vq, where dvp is the degree of vp. For i = 1, 2, ..., m, the maximum of ΙλiΙ as the DSt-spectral radius of ΓG and the sum of ΙλiΙ as DSt-energy of ΓG, where λi are the eigenvalues of DSt(ΓG). These notations can be applied analogously to the degree subtraction matrix of the commuting graph, DSt(ΓG). Throughout this paper, we provide DSt-spectral radius and DSt-energy of ΓG and ΓG for dihedral groups of order 2n, where n ≥ 3. We then present the correlation of the energies and their spectral radius. [ABSTRACT FROM AUTHOR]

A commuting graph is a graph denoted by C (G , X) where G is any group and X, a subset of a group G, is a set of vertices for C (G , X) . Two distinct vertices, x , y ∈ X , will be connected by an edge if the commutativity property is satisfied or x y = y x . This study presents results for the connectivity of C (G , X) when G is a symmetric group of degree n, Sym (n) , and X is a conjugacy class of elements of order three in G. [ABSTRACT FROM AUTHOR]