On Edge-MetaGraphs
Takaaki Fujita1,* Ajoy Kanti Das2 Suman Das3 Sankar Prasad Mondal4 Volkan Duran5
1 Independent Researcher, Tokyo, Japan
2 Associate Professor, Department of Mathematics, Tripura University, Agartala-799022, Tripura, India
3 Assistant Professor Grade II (Mathematics), Department of Education, National Institute of Technology Calicut, Kozhikode-673601,
Kerala, India
4 Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata-741249, West
Bengal, India
5 Department of Computer Engineering, Igdır University, Turkey
Emails: Takaaki.fujita060@gmail.com · ajoykantidas@gmail.com · dr.sumandas1995@gmail.com · sankar.mondal02@gmail.com ·
Received: October 30, 2025 Revised: December 26, 2025 Accepted: January 29, 2026 ⋆ Corresponding author
ABSTRACT
This paper studies graph-based higher-order structures related to metagraphs and edge-labeled hierarchical networks.
After reviewing MetaGraphs and Iterated MetaGraphs, we introduce the notion of an Edge-MetaGraph, in which each
edge is labeled by a two-ported internal graph, allowing edge-substitution expansion through port gluing. We then
define Iterated Edge-MetaGraphs recursively, so that edges may carry nested Edge-MetaGraph structures. Concrete
examples from biomedical systems, software pipelines, and logistics are presented to illustrate the expressive power
of the proposed framework.
Keywords: Edge-MetaGraph Iterated Edge-MetaGraph MetaGraph Iterated MetaGraph
1. PRELIMINARIES
This section establishes notation and summarizes the settheoretic
and combinatorial notions needed in the sequel.
Throughout, all sets that serve as vertex domains are assumed
to be finite, and we write N0 := N∪{0}.
1.1 MetaGraph (Graph of Graphs)
Graph theory investigates mathematical structures consisting
of vertices and edges for modeling relationships and connectivity
[1]. However, ordinary graphs may not always
provide sufficient flexibility for certain applications. To address
such limitations, various extensions of graph theory
have been developed, including Fuzzy Graphs [2, 3] and
Neutrosophic Graphs [4, 5, 6]. Another notable extension
is the MetaGraph. A MetaGraph is a graph whose vertices
are themselves graphs, with edges representing specified relations
between those graphs (cf. [7, 8, 9]). A MetaGraph
is also known as a Graph of Graphs (cf. [10, 11, 12]). Related
generalized graph structures capable of representing
higher-order or structurally complex relationships, such as
HyperGraphs[13, 14, 15] and SuperHyperGraphs[16, 17],
are also well known.
Definition 1.1 (Metagraph (graph of graphs)). [18] Fix a
nonempty universe G of finite graphs (undirected, loopless
by default) and a nonempty family of binary relations
R ⊆ P
G×G
.
A metagraph over (G,R) is a directed, labelled multigraph
M = (V,E, s, t,λ)
with
V ⊆ G, s, t : E →V, λ : E →R,