Let $G=(V,E)$ be a simple connected graph. A vertex $x\in V(G)$ resolves the elements $u,v\in E(G)\cup V(G)$ if $d_G(x,u)\ne d_G(x,v)$. A subset $S\subseteq V(G)$ is a mixed metric resolving set for $G$ if every two elements of $G$ are resolved by some vertex of $S$. A set of smallest cardinality of mixed metric generator for $G$ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $J_n≆P_n$ with $n\ge 3$ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph.

For a signature function $\mathrm{\Psi }:E(H)⟶\{\pm 1\}$ with underlying graph $H$, a signed graph (S.G) $\hat{H}=(H,\mathrm{\Psi })$ is a graph in which edges are assigned the signs using the signature function $\mathrm{\Psi }$. An S.G $\hat{H}$ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, $-\hat{h}(\hat{H})$ is also an eigenvalue of $\hat{H}$. A non singular S.G $\hat{H}$ is said to fulfill the property $(\mathcal{S}\mathcal{R})$ if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, its reciprocal is also an eigenvalue of $\hat{H}$ (with multiplicity as that of $\hat{h}(\hat{H})$). A non singular S.G $\hat{H}$ is said to fulfill the property $(-\mathcal{S}\mathcal{R})$ if for every eigenvalue $\hat{h}(\hat{H})$ of $\hat{H}$, its negative reciprocal is also an eigenvalue of $\hat{H}$ (with multiplicity as that of $\hat{h}(\hat{H})$). In this article, non bipartite unbalanced S.Gs ${\hat{\mathfrak{C}}}_3^{(m,1)}$ and ${\hat{\mathfrak{C}}}_5^{(m,2)}$, where $m$ is even positive integer have been constructed and it has been shown that these graphs fulfill the symmetric eigenvalue property, the S.Gs ${\hat{\mathfrak{C}}}_3^{(m,1)}$ also fulfill the properties $(-\mathcal{S}\mathcal{R})$ and $(\mathcal{S}\mathcal{R})$, whereas the S.Gs ${\hat{\mathfrak{C}}}_5^{(m,2)}$ are close to fulfill the properties $(-\mathcal{S}\mathcal{R})$ and $(\mathcal{S}\mathcal{R})$.

For a connected network $\mathrm{\Gamma }$, the distance between any two vertices is the length of the shortest path between them. A vertex $c$ in a connected network is said to resolve an edge $e$ if the distances of $c$ from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as $\eta :V(\mathrm{\Gamma })\to [0,1]$ such that $\eta (R_x(e))\ge 1$ for each edge $e\in E(\mathrm{\Gamma })$, where $R_x(e)$ represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as $di{m}_{LF}(\mathrm{\Gamma })=min\{|\eta |:\eta istheminimallocalresolvingfunctionof\mathrm{\Gamma }\},$ where $|\eta |=\sum _{a\in R_x(e)}\eta (a)$. In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity.