Characteristic polynomial and Wiener index of the compressed zero divisor graph \(\Gamma_{E}[\mathbb {Z}_{p^n}]\)
Serdica Mathematical Journal, Tome 46 (2020) no. 1, pp. 61-72
Cet article a éte moissonné depuis la source Bulgarian Digital Mathematics Library
The zero divisor graph of a commutative ring \(R\), denoted by \(\Gamma[R]\), is a graph whose vertices are non-zero zero divisors of \(R\) and two vertices are adjacent if their product is zero. The relation on \(R\) given by \(r\backsim s\) if and only if ${\rm ann}_R(r) = {\rm ann}_R(s)\) is an equivalence relation. The compressed zero divisor graph \(\Gamma_E(R)\) is the (undirected) graph whose vertices are the equivalence classes induced by \(\backsim\) other than \([0]\) and \([1]\), such that distinct vertices \([r]\) and \([s]\) are adjacent in \(\Gamma_E(R)\) if and only if \(rs = 0\). We show that the coefficients of the compressed zero divisor graph \(\Gamma_{E}[\mathbb{Z}_{p^n}]\) form a Pascal like triangle and thus derive the characteristic polynomial in terms of binomial coefficients. We also calculate the Wiener index of \(\Gamma_{E}[\mathbb{Z}_{p^n}]\).
Keywords:
compressed zero divisor graph, characteristic polynomial, binomial coefficients, Wiener index, 13A70, 05C12, 05C25, 13A15
@article{SMJ2_2020_46_1_a4,
author = {Reddy, Buruju and Jain, Rupali and Laxmikanth, Nandala},
title = {Characteristic polynomial and {Wiener} index of the compressed zero divisor graph {\(\Gamma_{E}[\mathbb} {{Z}_{p^n}]\)}},
journal = {Serdica Mathematical Journal},
pages = {61--72},
year = {2020},
volume = {46},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SMJ2_2020_46_1_a4/}
}
TY - JOUR
AU - Reddy, Buruju
AU - Jain, Rupali
AU - Laxmikanth, Nandala
TI - Characteristic polynomial and Wiener index of the compressed zero divisor graph \(\Gamma_{E}[\mathbb {Z}_{p^n}]\)
JO - Serdica Mathematical Journal
PY - 2020
SP - 61
EP - 72
VL - 46
IS - 1
UR - http://geodesic.mathdoc.fr/item/SMJ2_2020_46_1_a4/
LA - en
ID - SMJ2_2020_46_1_a4
ER -
%0 Journal Article
%A Reddy, Buruju
%A Jain, Rupali
%A Laxmikanth, Nandala
%T Characteristic polynomial and Wiener index of the compressed zero divisor graph \(\Gamma_{E}[\mathbb {Z}_{p^n}]\)
%J Serdica Mathematical Journal
%D 2020
%P 61-72
%V 46
%N 1
%U http://geodesic.mathdoc.fr/item/SMJ2_2020_46_1_a4/
%G en
%F SMJ2_2020_46_1_a4
Reddy, Buruju; Jain, Rupali; Laxmikanth, Nandala. Characteristic polynomial and Wiener index of the compressed zero divisor graph \(\Gamma_{E}[\mathbb {Z}_{p^n}]\). Serdica Mathematical Journal, Tome 46 (2020) no. 1, pp. 61-72. http://geodesic.mathdoc.fr/item/SMJ2_2020_46_1_a4/