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STRUCTURAL PROPERTIES OF THE GRAPHS ARISING FROM CONGRUENCES OVER SET OF MODULI
Keywords:
vertex degrees, graphs, congruences, set of moduli, Laplacian.DOI:
https://doi.org/10.17654/0972555525005Abstract
For any positive integer $n$ divisible by 3 , let $k$ divide $n$ and $M_k=\{k, 2 k\}$. Taking $M_k$ as the set of moduli, we define a graph with the vertices $0,1,2, \ldots, n-1$, and an edge between any two vertices $x$ and $y$ if and only if $x \equiv y(\bmod m)$ for $m \in M_k$. We label this graph as $G\left(n, M_k\right)$. We enumerate triangles and the number of components of the proposed graph. Further, we find the independence number, clique number, spectrum, Laplacian spectrum, energy and Laplacian energy of the proposed graph. Finally, we prove that the proposed graphs can be reduced to path graphs by substituting the moduli set.
Received: October 8, 2024
Accepted: November 22, 2024
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