## Mathematics Faculty Publications

Article

3-23-2021

#### Abstract

N be a vertex coloring of a simple; connected graph G. For a vertex v of G; the color sum of v; denoted by σ(G) is the sum of the colors of the neighbors of v. If σ(u)≠σ(v) for any two adjacent vertices u and v of G; then c is called a sigma coloring of G. The sigma chromatic number of G; denoted by σ(G); is the minimum number of colors required in a sigma coloring of G. Let max(c) be the largest color assigned to a vertex of G by a coloring c. The sigma value of G; denoted by υ(G); is the minimum value of max(c) over all sigma k-colorings c of G for which σ(G)= k. On the other hand; the sigma range of G; denoted by ρ(G); is the minimum value of max(c) over all sigma colorings c of G. In this paper; we determine the sigma value and the sigma range of the join of a finite number of even cycles of thesame order."}" data-sheets-userformat="{"2":12672,"10":2,"11":3,"15":"Calibri","16":11}" style="font-size: 11pt; font-family: Calibri, Arial;">Let c: V(G) -->N be a vertex coloring of a simple; connected graph G. For a vertex v of G; the color sum of v; denoted by σ(G) is the sum of the colors of the neighbors of v. If σ(u)≠σ(v) for any two adjacent vertices u and v of G; then c is called a sigma coloring of G. The sigma chromatic number of G; denoted by σ(G); is the minimum number of colors required in a sigma coloring of G. Let max(c) be the largest color assigned to a vertex of G by a coloring c. The sigma value of G; denoted by υ(G); is the minimum value of max(c) over all sigma k-colorings c of G for which σ(G)= k. On the other hand; the sigma range of G; denoted by ρ(G); is the minimum value of max(c) over all sigma colorings c of G. In this paper; we determine the sigma value and the sigma range of the join of a finite number of even cycles of the same order.

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