WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
The Reflexive Edge Strength of Cycles Plus One Edge
Authors: ,
Abstract: Let $$G$$ be a simple graph with a vertex set $$V (G)$$ and edge set $$E(G)$$. Given a vertex labeling $$fV$$: $$V (G)→ \lbrace0, 2, 4, ..., 2k_{v}\rbrace$$ and an edge labelings $$fE : E(G) → \lbrace1, 2, 3, ..., 2k_{e}\rbrace$$. Define a function $$f$$ by $$f(x) = fV (x)$$ if $$x ∈ V (G)$$ and $$f(x) = fE(x)$$ if $$x ∈ E(G)$$. We call $$f$$ be the total $$k-$$labeling where $$k = max\lbrace k_{e}, k_{v}\rbrace$$. A total $$k$$-labeling $$f$$ is called an edge irregular reflexive $$k$$-labeling of $$G$$ if every two distinct edge
$$xy$$ and $$x^{′}y^{′}$$, we have $$wt_{f} (xy) \neq wt_{f} (x′y′)$$ where $$wt_{f} (uv) = f(u) + f(uv) + f(v)$$ if uv is an edge of $$G$$. The reflexive edge strength of $$G$$, denoted by $$res(G)$$ is the minimum $$k$$ for $$G$$ which has an edge irregular reflexive $$k$$-labeling. In this paper, we give the exact value of $$res(C_{n} +e)$$ where $$C_{n} +e$$ is a cycle of order n plus one edge
which contains a triangle.
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Pages: 37-41
DOI: 10.37394/23206.2024.23.4