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Journal of Discrete Mathematical Sciences and Cryptography

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Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Laplacian energy and color based energy for graphs associated with commutative rings

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pp. 2431–2445Vol. 27Issue 8December 2024DOI: 10.47974/JDMSC-2016 Crossmark XML
Received:
17 Apr 2024
Published Online:
18 Dec 2024
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2016
Pages:
2431–2445

Abstract

Let’s assume that ℛ is a commutative ring and its prime graph is PG(ℛ). The vertices of this graph represent elements in ℛ, and an edge connects two different vertices (xa, yb) if and only if xa . yb = 0 or yb . xa = 0. The commutative ring R’s prime digraph is a straightforward graph whose vertices stand in for the equivalence classes of its members, where α ≡ β when anh(α) = anh(β). These equivalence classes are denoted by α–, where ≤ is defined such that α ≤ β when anh(α) ⸦ anh(β), and < on the equivalence classes such that α < β when anh(α)⸦ anh(β). An arc from β to α exists if α > 0; otherwise, an arc exists from α to β, where this directed graph is denoted as PDG(R). The Laplacian energy LPE(G) = ∑ni=1‍|μi – 2m/n| for a graph G, where μi represents eigenvalues of its Laplacian matrix. This paper discusses Laplacian energy for both directed prime graphs also color based energy for graphs.

Keywords

Commutative ringGraphsLaplacian energyColor based energy

Subject Classifications

05C2505C5005C75

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