A vertex <inline-formula> <tex-math notation="LaTeX">k\in V_{G} </tex-math></inline-formula> determined two elements (vertices or edges) <inline-formula> <tex-math notation="LaTeX">\ell,m \in ...V_{G}\cup E_{G} </tex-math></inline-formula>, if <inline-formula> <tex-math notation="LaTeX">d_{G}(k,\ell)\neq d_{G}(k,m) </tex-math></inline-formula>. A set <inline-formula> <tex-math notation="LaTeX">R_ {\text {m}} </tex-math></inline-formula> of vertices in a graph <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula> is a mixed metric generator for <inline-formula> <tex-math notation="LaTeX">G </tex-math></inline-formula>, if two distinct elements (vertices or edges) are determined by some vertex set of <inline-formula> <tex-math notation="LaTeX">R_ {\text {m}} </tex-math></inline-formula>. The least number of elements in the vertex set of <inline-formula> <tex-math notation="LaTeX">R_ {\text {m}} </tex-math></inline-formula> is known as mixed metric dimension, and denoted as <inline-formula> <tex-math notation="LaTeX">dim_{m}(G) </tex-math></inline-formula>. In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are <inline-formula> <tex-math notation="LaTeX">P^{2}_{n} </tex-math></inline-formula> the square of a path, <inline-formula> <tex-math notation="LaTeX">T(P_{n}) </tex-math></inline-formula> total graph of a path, the middle graph of a path <inline-formula> <tex-math notation="LaTeX">M(P_{n}) </tex-math></inline-formula>, and splitting graph of a path <inline-formula> <tex-math notation="LaTeX">S(P_{n}) </tex-math></inline-formula>. We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path <inline-formula> <tex-math notation="LaTeX">S(P_{n}) </tex-math></inline-formula>.
The p-type networks are designed with the help of CVNET at topo group Cluj and also given support by nano studio. Such networks develop new p-type surfaces and also represent the decorations of the ...surfaces. This patched network is designed by two repeated units. The first one is triphenylene having a Z-pen formula and the second one is triphenylene with A-phe. Furthermore, these decorations are acquired as the result of map operations represented in the CVNET software, while its assembling is conducted with the help of the nano studio program. In the literature, its topology is discussed by Omega polynomials which is an applied graph theory topic. Another most applied topic of graph theory is known as the resolvability parameter. So this article studied the resolvability parameters of patched networks, such as metric dimension, and edge metric dimension. These parameters are defined as a resolving set is a subset of vertices of a graph with a condition that each vertex of that graph has a unique code or representation with respect to the chosen subset. Its minimum cardinality is known as metric dimension, while the edge metric dimension is defined by the minimum count of members in the edge resolving set and this set is defined as according to a chosen subset each edge of a graph has unique representations, then this set is known as edge resolving set. A resolving set is a subset of vertices of a graph with a condition that each edge of that graph has a unique code or representation with respect to the chosen subset. It is minimum cardinality is known as the edge metric dimension.
A vertex <inline-formula> <tex-math notation="LaTeX">$k\in V_{G}$ </tex-math></inline-formula> determined two elements (vertices or edges) <inline-formula> <tex-math notation="LaTeX">$\ell,m \in ...V_{G}\cup E_{G}$ </tex-math></inline-formula>, if <inline-formula> <tex-math notation="LaTeX">$d_{G}(k,\ell)\neq d_{G}(k,m)$ </tex-math></inline-formula>. A set <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> of vertices in a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is a mixed metric generator for <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, if two distinct elements (vertices or edges) are determined by some vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula>. The least number of elements in the vertex set of <inline-formula> <tex-math notation="LaTeX">$R_ {\text {m}}$ </tex-math></inline-formula> is known as mixed metric dimension, and denoted as <inline-formula> <tex-math notation="LaTeX">$dim_{m}(G)$ </tex-math></inline-formula>. In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are <inline-formula> <tex-math notation="LaTeX">$P^{2}_{n}$ </tex-math></inline-formula> the square of a path, <inline-formula> <tex-math notation="LaTeX">$T(P_{n})$ </tex-math></inline-formula> total graph of a path, the middle graph of a path <inline-formula> <tex-math notation="LaTeX">$M(P_{n})$ </tex-math></inline-formula>, and splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>. We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path <inline-formula> <tex-math notation="LaTeX">$S(P_{n})$ </tex-math></inline-formula>.
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