The Rainbow Mean Coloring of Some Operations of Graphs

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DOI: https://doi.org/10.28924/2291-8639-23-2025-304
Published: Nov 28, 2025
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K. Maheswari, G. Rajasekaran, The Rainbow Mean Coloring of Some Operations of Graphs, Int. J. Anal. Appl., 23 (2025), 304.

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K. Maheswari, G. Rajasekaran

Abstract

In a connected graph G with a minimum of three vertices, an edge coloring c allocates positive numbers to the edges. The chromatic mean of a vertex v is calculated by averaging the colors of all incident edges, provided that the result remains a positive integer. A coloring c is a rainbow mean coloring if each vertex in G has a unique chromatic mean. The rainbow mean index of c is the highest chromatic mean assigned to any vertex, while the rainbow mean index of G is the smallest possible maximum chromatic mean for all valid rainbow mean colorings. This study calculates the rainbow mean index of tensor product graphs, specifically G1 × G2, where G1 ∈ {Cq, Kq} and G2 ∈ {Ct, Kt}; Pq × H, where H ∈ {Ct, Kt, Wt, Ft} and χrm(H) = t. We also compute the rainbow mean index for the rooted product of two graphs, the join of two graphs, and the caterpillar graph.

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