Injective coloring of some graph operations

An injective coloring of a graph G is a vertex coloring such that any two vertices with a common vertex receive distinct colors. The injective chromatic number χi(G) of a graph G is the least k such that there is an injective k-coloring. Graph operations are important methods for constructing new graphs, and they play key roles in the design and analysis of networks. In this study, we give some sharp bounds (or exact values) of graph operations, including the Cartesian product, direct product, lexicographic product, union, join, and disjunction of graphs.