Mean coloring is an edge coloring c of a connected graph G of order 3 or more with positive integers if the chromatic mean of all vertices v of G are integers. The chromatic mean of a vertex v of G is given by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$cm(v)=\frac{\sum _{e \in E_{v}} c(e)}{deg(v)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mo>∑</mo> <mrow> <mi>e</mi> <mo>∈</mo> <msub> <mi>E</mi> <mi>v</mi> </msub> </mrow> </msub> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>e</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> where c(e) is the integers(colors) given to the edges incident to the vertex v. If each vertex has a distinct chromatic mean, then the edge coloring c is called a rainbow mean coloring. For a rainbow mean coloring c of a graph G, the maximum vertex color is the rainbow chromatic mean index, rm(c) of c. The rainbow mean index rm(G) of the graph G is defined as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$rm(G)=min \{rm(c): c\text { is the rainbow mean coloring} \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo stretchy="false">{</mo> <mi>r</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>c</mi> <mspace width="0.333333em" /> <mtext>is the rainbow mean coloring</mtext> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we determine the rainbow mean index of bistars and tadpole graphs. Also, we show that the rainbow mean index is same as the order for some derived graphs.

Rainbow Mean Index of Certain Derived Graphs

期刊论文
发表日期：

2026年8月4日

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Diya Jaji,
Tabitha Agnes Mangam

摘要

Mean coloring is an edge coloring c of a connected graph G of order 3 or more with positive integers if the chromatic mean of all vertices v of G are integers. The chromatic mean of a vertex v of G is given by $cm(v)=\frac{\sum _{e \in E_{v}} c(e)}{deg(v)}$ $c m (v) = \frac{\sum_{e \in E_{v}} c (e)}{d e g (v)}$ where c(e) is the integers(colors) given to the edges incident to the vertex v. If each vertex has a distinct chromatic mean, then the edge coloring c is called a rainbow mean coloring. For a rainbow mean coloring c of a graph G, the maximum vertex color is the rainbow chromatic mean index, rm(c) of c. The rainbow mean index rm(G) of the graph G is defined as $rm(G)=min \{rm(c): c\text { is the rainbow mean coloring} \}$ $r m (G) = m i n {r m (c) : c is the rainbow mean coloring}$ . In this paper, we determine the rainbow mean index of bistars and tadpole graphs. Also, we show that the rainbow mean index is same as the order for some derived graphs.