VOLUME 19 NUMBER 2 (July to December 2026)

Sigma values of some graph families under join and corona operations

Let c: V(H)→ Z^+ be a vertex coloring of a simple nontrivial connected graph H. The color sum of a vertex v∈ V(H),, denoted by σ(v), is defined as the sum of the colors of the vertices adjacent to v. A coloring c of H is called a sigma coloring if σ(u)≠σ(v) for any two adjacent vertices u and v of H. The sigma chromatic number of H, denoted by σ(H), is the minimum number of colors required in a sigma coloring of H. Let max⁡〖(c)〗 be the largest color assigned to a vertex of H by a coloring c. The sigma value of H, denoted by ν(H), is the minimum value of max⁡〖(c)〗 over all sigma l-colorings c of H for which σ(H) =l. In this paper, we determine the sigma value of the join of paths or cycles and the corona of a cycle with a star graph. We also generalize results on the sigma value of the corona of a path with another graph whose sigma value is at least 2, or the corona of a cycle with another graph whose sigma value is at least 2.