ON INCLUSIVE TOTAL DISTANCE IRREGULARITY STRENGTH OF JOINT PRODUCT GRAPHS

Abstract

Graph theory is one of the branches of mathematics that is rapidly developing due to its applications in solving various problems, including electronic networks, communication network models, transportation systems, and carbon reserve networks. The topology of these networks is simply represented using the concept of graphs. Specifically, graph labeling is widely used to address issues such as radio frequency assignment, computer network coding, data transfer optimization, and marketing distribution. Thus, conducting research to develop graph labeling methods is highly significant. Let $G=\left(V_G,E_G\right)$, be a simple connected graph, and $\lambda\ :V_G\cup E_G\rightarrow{1,\ 2,\ \ldots,\ k}$ be a labeling function on $G$. The inclusive weight of a vertex $v\in G$ is defined as the sum of the labels of $v$, all vertices in the $v$ neighborhood, and its incident edges. If all vertices in $V_G$ have a distinct inclusive weight, then $\lambda$ is called an inclusive distance vertex irregular total $k$-labeling of $G$ . The total distance vertex irregularity strength of $G$, denoted by $\widehat{tdis}\left(G\right)$, is the minimum $k$ for which such a labeling exists. This paper investigates the inclusive distance vertex irregular total $k$-labeling for certain classes of joint product graphs. Specifically, we determine the inclusive total distance irregularity strength of the joint product of path, cycle, and complete graphs, providing new insights into their structural labeling properties