Irregularity Index of Graph Transformations

Using graph transformations, a large number of graphs can be generated from smaller graphs. Graph products are considered among the fundamental concepts in graph theory. These products encompass a wide range of types, each of which produces new graphs with different properties and applications. Moreover, for an arbitrary graph 𝐺 with 𝑛 vertices and 𝑚 edges, the structure 𝑅𝑘(𝐺) produces a graph consisting of 𝑘 copies of 𝐺 with 𝑘𝑛 vertices and 𝑘2𝑚 m edges. The irregularity index of a connected graph 𝐺 is defined as 𝐼𝑟𝑟(𝐺)=Σ|𝑑𝐺(𝑢)−𝑑𝐺(𝑣)|𝑢𝑣∈𝐸(𝐺). In this paper, we compute the irregularity index for several graph products, including the lexicographic product, the corona product, and the direct product of graphs, and we establish bounds for the irregularity of these products. Additionally, we determine the irregularity of the structure 𝑅𝑘(𝐺) for an arbitrary graph 𝐺.