2023/04/06 by K, Najiya V, A, Chithra, Palanivel, Naveen
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.02905
The unitary addition Cayley graph Gn, n∈ Z+ is the graph whose vertex set is Zn, the ring of integers modulo n and two vertices u and v are adjacent if and only if u + v ∈ ∪n where ∪n is the set of all units of the ring. The Aα-matrix of a graph G is defined as Aα(G) = αD(G) + (1-α)A(G), α∈ [0, 1], where D(G) is the diagonal matrix of vertex degrees and A(G) is the adjacency matrix of G. In this paper, we investigate the Aα-eigenvalues for unitary addition Cayley graph and its complement. We determine bounds for Aα-eigenvalues of unitary addition Cayley graph when its order is odd. Consequently, we compute the Aα-energy of both Gn and its complement, Gn, for n=pm where p is a prime number and n even. Moreover, we obtain some bounds for energies of Gn and Gn when n is odd. We also define Aα-borderenergetic and Aα-hyperenergetic graphs and observe some classes for each.