WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
The Prime Graphs PG3(R) and PG4(R) over a Ring R
Authors: ,
Abstract: Let G be a simple graph. L(G) is the Laplacian matrix of G. We define a simple undirected graph $$PG_3(R)$$ whose vertices are all the elements of the ring R and two distinct vertices a, b are adjacent if and only if $$a.b=0$$ or $$b.a=0$$ or $$a+b=0$$ or $$a+b$$ is a unit element of R. Also, we define a simple undirected graph $$PG_4(R)$$ whose vertices are all the elements of the ring R and two distinct vertices a,b are adjacent if and only if $$a.b=0$$ or $$b.a=0$$ or $$a+b=0$$. In this paper we discuss degree of the vertices $$PG_{3}(R), PG_{4}(R)$$ for $$R=Z_n$$ where, $$Z_n$$ is the group of integer modulo n. Also, discuss planarity of the graph $$PG_{3}(R), PG_{4}(R)$$ for $$R=Z_n.$$ Here we introduced Laplacian of the graphs $$PG_{3}(R), PG_{4}(R)$$ for $$R=Z_p$$ and $$R=Z{_p}$$ x $$Z{_p} $$ where, p is prime and we find their girth, algebraic connectivity, clique number and discuss Eulerian property.
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Pages: 171-183
DOI: 10.37394/23206.2023.22.22