On the structure and spectra of an induced subgraph of essential ideal graph of ℤ_n

2023/10/17 by Jamsheena, P., Chithra, A. V.
#05C25 #05C40 #05C50 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2310.10999

Abstract

Let R be a commutative ring with unity. The essential ideal graph E_R of R is a graph in which the vertex set comprises of set of all nonzero proper ideals of R and two vertices I and K are adjacent if and only if I+K is an essential ideal. In this paper, we discuss the structure of an induced subgraph of the essential ideal graph of the ring ℤ_n as a \mathscrG-generalized join graph and thereby completely determine the structure of E_ℤ_n. Also, we prove a characterization of E_ℤ_n to be Laplacian integral in terms of the vertex-weighted Laplacian matrix of annihilating ideal graph of ℤ_n for n= ∏_i=1^k p_i. Further, we discuss the eigenvalues of various matrices like adjacency matrix, Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the induced subgraph of the essential ideal graph of ℤ_n. Finally, we obtain the upper bounds of spectral radius and algebraic connectivity of E_ℤ_n and compute the values of n for which these bounds are attained.

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