Advances and Applications in Discrete Mathematics
Volume 32, , Pages 63 - 89
(July 2022) http://dx.doi.org/10.17654/0974165822033 |
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STUDY OF SOME GRAPHICAL PARAMETERS OF SOME GRAPH STRUCTURE
Pinku Sarkar and Kuntala Patra
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Abstract: Let G be a simple graph. is the Laplacian matrix of G and is the algebraic connectivity of the graph G. Pawar and Joshi [9] defined a simple undirected graph on a ring R having vertex set R and any two distinct vertices a, b are adjacent if and only if or or is a unit element of R. Satyanarayana et al. [11] defined prime graph by taking all elements of the ring R as vertices and two distinct vertices a, b are adjacent if and only if or Another simple undirected graph ℾ2 is defined by Gupta [10] whose vertices are all the non-zero elements of the ring R and two distinct vertices a, b are adjacent if and only if or or is a zero divisor (including zero). In this paper, we prove that for any prime p, the graphs and are planar if and only if Also, we prove that the graphs ℾ2 and are not Eulerian for any prime p. Here we discuss Laplacian and algebraic connectivity of ℾ2 ℾ2 and where p is a prime. We also find their girth, vertex connectivity and discuss planarity and Eulerian properties.
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Keywords and phrases: Laplacian matrix, algebraic connectivity, prime graph PG(R) and PG1(R) of a ring, planarity |
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