Meta-Fuzzy Graph, Meta-Neutrosophic Graph, Meta-Digraph, and Meta-MultiGraph with some applications

Graph theory investigates mathematical structures consisting of vertices and edges to model relationships and connectivity [1, 2]. A MetaGraph is a higher-level graph whose vertices are themselves graphs, with edges representing specified relations among those graphs. An Iterated MetaGraph extends this idea recursively: its vertices are MetaGraphs, yielding a hierarchy of graph-of-graphs structures across multiple levels. Fuzzy graphs incorporate fuzzy membership functions on vertices and edges, thereby capturing uncertainty and graded strength of connectivity. Neutrosophic graphs generalize this further by assigning to each vertex and edge three independent membership values—truth, indeterminacy, and falsity—providing a more comprehensive framework for uncertainty. A weighted graph is a graph in which each edge is assigned a numerical value (weight), typically representing cost, distance, or intensity. Multigraphs, which allow multiple parallel edges and loops, appear naturally when such multiplicities are required. Bidirected graphs (bidigraphs) assign local orientations to each vertex-edge incidence, allowing edges to point independently at both ends. In this paper, we extend the frameworks of fuzzy graphs, neutrosophic graphs, multigraphs, weighted graphs, digraphs, and bidirected graphs by embedding them into the unified setting of MetaGraphs and Iterated MetaGraphs.