Graph Theoretical Properties of Fibonacci and Generalized Cubes
Summary
Fibonacci cubes and their generalizations form families of subgraphs of the n‐dimensional hypercube obtained by excluding vertices whose binary labels contain prescribed patterns. These graphs exhibit rich structural and metric characteristics, including precise connectivity and edge‐connectivity values, isometric embedding properties, Hamiltonicity and fault‐tolerant routing potential. Enumeration of k‐dimensional subcubes via cube polynomials yields closed‐form generating functions, while analyses of diameter, radius and periphery inform performance bounds in interconnection networks. Applications span distributed computing topologies, chemical graph models and combinatorial optimisation, with recent work unveiling deeper algebraic and metric interrelations among Fibonacci, Lucas, Padovan and further sequence‐defined cubes.
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Graph Theoretical Properties of Fibonacci and Generalized Cubes publication trend
The graph below shows the total number of articles in graph theoretical properties of fibonacci and generalized cubes across all publications each year (not limited to Nature Index journals).
Technical terms
Hypercube: An n-dimensional graph whose vertices are binary n-tuples, with edges joining vertices differing in exactly one coordinate.
Fibonacci cube: The subgraph of an n-cube induced by vertices whose binary labels contain no two consecutive 1s.
Generalized Fibonacci cube: A subgraph of an n-cube formed by removing all vertices whose labels contain a specified binary pattern as a factor.
Lucas cube: A variant of the Fibonacci cube defined by exclusion of binary strings derived from Lucas sequence patterns.
Cube polynomial: A generating function whose coefficient of x^k counts the number of k-dimensional hypercube subgraphs within a given graph.
Connectivity: The minimum number of vertices (or edges) whose removal disconnects a graph.
Diameter: The maximum distance between any two vertices in a graph, measured by the length of a shortest path.
References
- Connectivity of Fibonacci cubes, Lucas cubes and generalized cubes. Discrete Mathematics & Theoretical Computer Science (2015).
- On the Cube Polynomials of Padovan and Lucas–Padovan Cubes. Symmetry (2023).
- Horadam–Lucas Cubes. Axioms (2024).
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