Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle

Abstract

Edge-counting vectors of subgraphs of Cartesian products are introduced as the counting vectors of the edges that project onto the factors. For several standard constructions their edge-counting vectors are computed. It is proved that the edge-counting vectors of Fibonacci cubes are precisely the rows of the Fibonacci triangle and that the edge-counting vectors of Lucas cubes are Fn−1-constant vectors. Some problems are listed along the way.

Binary Coding of Resonance Graphs of Catacondensed Polyhexes

“…It turned out that the resonance graphs are associated, somewhat surprisingly, with some other well-known families of graphs. A well-known example are Fibonacci cubes, a class of graphs used in network design, which are precisely the resonance graphs of zigzag benzenoid chains (also known as fibonaccenes) [14]. Later, all plane bipartite graphs with perfect matchings whose resonance graphs are Fibonacci cubes were determined [26].…”

Outerplane Bipartite Graphs with Isomorphic Resonance Graphs

“…There are outerplane bipartite graphs G and G ′ whose inner duals are isomorphic paths but with nonisomorphic resonance graphs. For example, let G be a linear benzenoid chain (a chain in which every non-terminal hexagon is linear) with n hexagons, and let G ′ be a fibonaccene (a benzenoid chain in which every non-terminal hexagon is angular, see [12]) with n hexagons, where n > 2. Then the inner dual T of graph G is isomorphic to the inner dual T ′ of graph G ′ , since T and T ′ are both paths on n vertices.…”

A Class of Fibonacci Matrices, Graphs, and Games

“…These are similar to the hypercube graphs, but having a Fibonacci number of vertices. Applications of these graphs have been found in parallel computing [17], routing and broadcasting in distributed computations [21] and chemical graph theory [22]. Another recent work introducing a class of Fibonacci graphs is [23].…”