Abstract
Let R be a commutative ring with identity. We consider ΓB(R) an annihilator graph of the commutative ring R. In this paper, we find Hosoya polynomial, Wiener index, Coloring, and Planar annihilator graph of Zn denote ΓB(Zn) , with n= pm or n=pmq, where p, q are distinct prime numbers and m is an integer with m ≥ 1 .
Highlights
Let R be a commutative ring with identity the annihilator of R is the set of all element x ∈ R satisfy ann(R) = {x ∈ R: x. y = 0, ∀ y ∈ R} [6], and let ann(R) be the set of all annihilator in R
The chromatic number of vertices the graph ΓB(Zpmq) is p 2 + 2 (when m is an odd)
Summary
P 2 = 0) be the product pm or one of its complications of the number pm which is equal to (0) in the ring Zpm. the graph ΓB(Zpm) contains a subgraph homeomorphic complete bipartite graph k m+2 m−2 m−2 it is the largest (p 2 − p 2 ),p 2 complete bipartite graph there is in the graph ΓB(Zpm) as in the Figure (2.4). ΓB(Zpm) when m is an odd it does not contain subgraph homeomorphic k5 or k3,3 it is planar graphs by kuratowski’s Theorem.
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