Structural Properties and Characteristic Polynomials of Cubic Power Graph of Dihedral Group

Cubic power graph of dihedral group <img src=image/13440420_01.gif> having order <img src=image/13440420_02.gif> and identity element <img src=image/13440420_03.gif>, <img src=image/13440420_04.gif> is a finite, simple, undirected graph for which two different vertices <img src=image/13440420_05.gif> are adjacent if and only if <img src=image/13440420_06.gif> or <img src=image/13440420_07.gif> for some <img src=image/13440420_08.gif> with <img src=image/13440420_09.gif>. In this research, conditions on <img src=image/13440420_12.gif> under which <img src=image/13440420_04.gif> is planar, are calculated. Also conditions on <img src=image/13440420_12.gif> under which <img src=image/13440420_04.gif> is Hamiltonian, are calculated. It is also shown that <img src=image/13440420_04.gif> is not an Eulerian graph. A polynomial that counts how many different ways there are to color a graph with a specific number of colors is called a chromatic polynomial. We have calculated the chromatic polynomial of <img src=image/13440420_04.gif> when <img src=image/13440420_10.gif>. Also chromatic polynomial of almost complete graph obtained by deleting <img src=image/13440420_11.gif> number of non-adjacent edges is obtained. Spectral graph theory is a branch of mathematics that examines a graph's characteristics in relation to its characteristic polynomial, eigenvalues, and eigenvectors of related matrices, such as its adjacency matrix, Laplacian matrix or degree based matrices. Characteristic polynomials of degree based matrices such as Laplacian matrix, Signless Laplacian matrix, Maximum matrix, Minimum matrix and Greatest common divisor degree matrix of <img src=image/13440420_04.gif> when <img src=image/13440420_10.gif>.