The adjacency spectrum and metric dimension of an induced subgraph of comaximal graph of ℤn.

Let R be a commutative ring with unity, and let Γ (R) denote the comaximal graph of R. The comaximal graph Γ (R) has vertex set as R, and any two distinct vertices x, y of Γ (R) are adjacent if R x + R y = R. Let Γ 2 (R) denote the induced subgraph of Γ (R) on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of Γ 2 (R) are adjacent if R x + R y = R. In this paper, we study the graphical structure as well the adjacency spectrum of Γ 2 (ℤ n) , where n ≥ 4 is a non-prime positive integer, and ℤ n is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of Γ 2 (ℤ n) are 0 with multiplicity n − φ (n) − D − 1 , and remaining eigenvalues are contained in the spectrum of a symmetric D × D matrix. We further calculate the rank and nullity of Γ 2 (ℤ n). We also determine all the eigenvalues of Γ 2 (ℤ n) whenever Γ 2 (ℤ n) is a bipartite graph. Finally, apart from determining certain structural properties of Γ 2 (ℤ n) , we conclude the paper by determining the metric dimension of Γ 2 (ℤ n). [ABSTRACT FROM AUTHOR]