Intertwining local (adjacency) metric dimension with the clique number of a graph

Abstract

Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ \omega(G)$, where $G\not\cong K_{n(G)}$ and $\omega(G)\geq3$. It is proved that $ {\rm dim}_{A,l}(G) \leq \left\lfloor \left(\frac{\omega(G) - 2}{\omega(G) - 1}\right)n(G)\right\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities.