Fibonacci Graphs and Their Constructions

• Fibonacci Graphs are a diverse family of graphs whose construction is governed by Fibonacci or Lucas recurrences, encompassing directed acyclic graphs, Fibonacci cubes, and additive models.
• They encode Fibonacci behavior through constrained binary strings, path enumeration, and recursive invariants, offering deep combinatorial and geometric insights.
• Applications span from algebraic expression optimization in directed st-dags to precise metric analyses in hypercube-induced graphs and extremal graph theory in additive constructions.

Searching arXiv for recent and foundational papers on Fibonacci graphs to ground the article. “Fibonacci graphs” denotes several non-equivalent graph-theoretic constructions whose defining combinatorics are governed by Fibonacci or Lucas recurrences. In the literature summarized here, the term covers at least four major settings: directed acyclic graphs with edges $(v,v+1)$ and $(v,v+2)$; induced subgraphs of hypercubes such as Fibonacci cubes, Lucas cubes, Fibonacci-run graphs, and Lucas-run graphs; additive graphs such as Fibonacci-sum graphs; and order-theoretic structures such as the Young–Fibonacci graph. A plausible unifying theme is that these families encode Fibonacci behavior in one of three ways: through constrained binary strings, through path or walk enumeration, or through invariants whose extremal values are controlled by Fibonacci-number recurrences (Korenblit et al., 2013, Klavzar et al., 2013, Eğecioğlu et al., 2020).

1. Terminological scope and canonical families

The terminology is not uniform. In one established usage, a Fibonacci graph is a directed acyclic graph on vertices $\{1,2,\dots,n\}$ with edges $(v,v+1)$ and $(v,v+2)$; in another, the phrase refers to hypercube-induced families such as Fibonacci cubes; elsewhere it appears in compounds such as Fibonacci-run graph, Fibonacci-sum graph, Young–Fibonacci graph, and $k$-bonacci graph (Korenblit et al., 2013, Klavzar et al., 2013, Eğecioğlu et al., 2020, Arman et al., 2017, Evtushevsky, 2020, Kirgizov et al., 2022).

Family	Defining object	Fibonacci feature
Directed Fibonacci graph	DAG on $\{1,\dots,n\}$ with $(v,v+1)$, $(v,v+2)$	Source–sink path structure and expression recurrences
Fibonacci cube $\Gamma_n$	Induced subgraph of $(v,v+2)$0 on strings with no consecutive $(v,v+2)$1s	$(v,v+2)$2
Fibonacci-run graph $(v,v+2)$3	Induced subgraph on run-constrained strings	$(v,v+2)$4
Lucas cube / Lucas-run graph	Cyclic versions of the preceding cube families	Lucas-type cyclic restriction
Fibonacci-sum graph $(v,v+2)$5	Vertices $(v,v+2)$6, edges when $(v,v+2)$7 is Fibonacci	Adjacency determined by Fibonacci numbers
Young–Fibonacci graph	Hasse diagram on words over $(v,v+2)$8	Rank sizes are Fibonacci-counted

This multiplicity of meanings is substantive rather than merely terminological. The directed acyclic Fibonacci graph is primarily a test case for algebraic-expression minimization in non-series-parallel st-dags, whereas the hypercube-induced families are partial-cube or daisy-cube objects studied via embeddings, cube polynomials, and distance distributions. The additive and order-theoretic families, by contrast, organize Fibonacci behavior through edge predicates or graded-poset path counting (Korenblit et al., 2013, 0903.2507, Arman et al., 2017, Evtushevsky, 2020).

2. Hypercube-induced families

The classical Fibonacci cube $(v,v+2)$9 is the induced subgraph of the hypercube $\{1,2,\dots,n\}$0 on binary strings $\{1,2,\dots,n\}$1 with $\{1,2,\dots,n\}$2 for all $\{1,2,\dots,n\}$3. The Lucas cube is the corresponding cyclic variant with the extra restriction $\{1,2,\dots,n\}$4. These families are canonical because their vertex sets are Fibonacci- and Lucas-counted constrained words, and because they retain enough hypercube structure to support metric, enumerative, and asymptotic analysis (Klavzar et al., 2013).

The Fibonacci-run graphs $\{1,2,\dots,n\}$5 replace the “no consecutive $\{1,2,\dots,n\}$6s” condition by a run-length constraint: every run of $\{1,2,\dots,n\}$7s is immediately followed by a strictly longer run of $\{1,2,\dots,n\}$8s. The permitted blocks form the monoid

$\{1,2,\dots,n\}$9

that is, blocks $(v,v+1)$0 with $(v,v+1)$1. The vertex set is described by

$(v,v+1)$2

and $(v,v+1)$3 is an induced subgraph of $(v,v+1)$4. A bijection $(v,v+1)$5 from extended Fibonacci strings to run-constrained strings,

$(v,v+1)$6

extended multiplicatively, yields

$(v,v+1)$7

Thus Fibonacci-run graphs preserve the Fibonacci-number vertex count of Fibonacci cubes while changing the admissible language and the adjacency geometry (Eğecioğlu et al., 2020, Mollard, 2024).

This change is structurally significant. For $(v,v+1)$8,

$(v,v+1)$9

so Fibonacci-run graphs have fewer edges than Fibonacci cubes. Their asymptotic average degree satisfies

$(v,v+2)$0

whereas for Fibonacci cubes and Lucas cubes,

$(v,v+2)$1

Likewise, Fibonacci-run graphs are partial cubes only for $(v,v+2)$2, and they are not median graphs for $(v,v+2)$3, unlike Fibonacci cubes (Eğecioğlu et al., 2020, Klavzar et al., 2013).

The Lucas-run graphs $(v,v+2)$4 are the cyclic version of Fibonacci-run graphs. A string is circular-run-constrained if every run of $(v,v+2)$5s is immediately followed circularly by a strictly longer run of $(v,v+2)$6s, and

$(v,v+2)$7

They form induced subgraphs of $(v,v+2)$8, and the construction mirrors the Fibonacci cube/Lucas cube relation in the run-constrained setting (Mollard, 2024).

A more recent generalization is given by Munarini graphs $(v,v+2)$9. They are built from generalized Pell strings over $k$0 with no runs of $k$1s of odd length, and satisfy

$k$2

Their significance is that, unlike generalized Pell graphs for $k$3, they remain daisy cubes. The vertex count is

$k$4

with $k$5, and the family inherits recursive structure, cube polynomials, and maximal cube polynomials analogous to those of Fibonacci cubes (Mollard, 14 May 2026).

3. Directed acyclic Fibonacci graphs and algebraic expressions

In the algebraic-expression literature, a Fibonacci graph $k$6 is the st-dag on vertices $k$7 with edge set

$k$8

The edges $k$9 and $\{1,\dots,n\}$0 are labeled $\{1,\dots,n\}$1 and $\{1,\dots,n\}$2, respectively. This family is a generic non-series-parallel st-dag: it contains the forbidden pattern that obstructs series-parallel reduction, so its source–sink expression cannot in general be reduced to a read-once formula (Korenblit et al., 2013, Korenblit et al., 2013).

The canonical expression is the sum of the products of edge labels along all source-to-sink paths. Two complexity measures are standard: $\{1,\dots,n\}$3 The sequential-paths method writes the canonical sum directly and yields Fibonacci-counted path growth: $\{1,\dots,n\}$4 For $\{1,\dots,n\}$5, this method produces an expression with 201 terms and 33 plus operators (Korenblit et al., 2013).

The decisive improvement is the decomposition method. For a subgraph with source $\{1,\dots,n\}$6 and sink $\{1,\dots,n\}$7, and a decomposition vertex $\{1,\dots,n\}$8 with $\{1,\dots,n\}$9,

$(v,v+1)$0

with base cases

$(v,v+1)$1

The path split is exact: every path from $(v,v+1)$2 to $(v,v+1)$3 either passes through $(v,v+1)$4 or bypasses it via $(v,v+1)$5. The optimal choice is the middle vertex of the interval, or one of the two middle vertices when the interval length is even. Within the decomposition framework, this choice minimizes both $(v,v+1)$6 and $(v,v+1)$7 (Korenblit et al., 2013, Korenblit et al., 2013).

This balanced recursion reduces the expression size from exponential to polynomial. The optimal decomposition method satisfies

$(v,v+1)$8

For a $(v,v+1)$9-vertex Fibonacci graph, the optimized expression has 31 terms and 11 plus operators, a large reduction from the sequential-paths form. The generalized decomposition method, which splits into $(v,v+2)$0 parts at each step, has

$(v,v+2)$1

and among uniform GD methods the minimum asymptotic complexity occurs at $(v,v+2)$2, namely the ordinary decomposition method (Korenblit et al., 2013, Korenblit et al., 2013).

The optimality question remains open at the global level. The papers formulate the conjecture that the optimal decomposition method gives an optimal representation for both complexity measures, and ask whether it is the only method that yields an optimal representation with respect to total term count (Korenblit et al., 2013).

4. Metric, cube-polynomial, and extremal invariants

A major line of work studies Fibonacci-type hypercube subgraphs through cube polynomials and distance cube polynomials. For a graph $(v,v+2)$3 containing $(v,v+2)$4,

$(v,v+2)$5

where $(v,v+2)$6 counts induced subgraphs isomorphic to $(v,v+2)$7 whose bottom vertex is at distance $(v,v+2)$8 from $(v,v+2)$9. For Fibonacci-run graphs, if a vertex $\Gamma_n$0 has Hamming weight $\Gamma_n$1 and down-degree $\Gamma_n$2, then the $\Gamma_n$3 having $\Gamma_n$4 as top vertex contribute

$\Gamma_n$5

This yields

$\Gamma_n$6

where $\Gamma_n$7 is the down-degree co-weight polynomial. A previously published generating function for $\Gamma_n$8 was shown to be erroneous because it omitted the factor $\Gamma_n$9; the corrected generating function is

$(v,v+2)$00

The same paper proves, in a stronger form, the conjectured recursive link between Lucas-run and Fibonacci-run graphs at the level of distance cube polynomials (Mollard, 2024).

For Fibonacci cubes and Lucas cubes, metric asymptotics are explicit. The asymptotic average eccentricity is

$(v,v+2)$01

and the asymptotic average degree is

$(v,v+2)$02

The same work introduces the hypercube density

$(v,v+2)$03

and proves

$(v,v+2)$04

It also establishes a correspondence between eccentricity in $(v,v+2)$05 and leaf depth in a suitably labeled Fibonacci tree (Klavzar et al., 2013).

A different invariant is the Fibonacci dimension $(v,v+2)$06, the smallest integer $(v,v+2)$07 such that $(v,v+2)$08 admits an isometric embedding into the Fibonacci cube $(v,v+2)$09. For a partial cube $(v,v+2)$10 with isometric dimension $(v,v+2)$11, the central theorem is

$(v,v+2)$12

where $(v,v+2)$13 is the auxiliary graph of semicubes and $(v,v+2)$14 is the minimum number of coordinating paths. The invariant satisfies

$(v,v+2)$15

and computing it is difficult: it is NP-complete to decide whether $(v,v+2)$16, NP-hard to approximate within $(v,v+2)$17, and nevertheless admits a $(v,v+2)$18-approximation algorithm in the general case (0903.2507).

The expression Fibonacci index refers to another invariant: $(v,v+2)$19 including the empty set. For fixed order $(v,v+2)$20 and stability number $(v,v+2)$21, Turán graphs uniquely maximize $(v,v+2)$22 among all graphs in $(v,v+2)$23, and Turán-connected graphs maximize it among connected graphs in $(v,v+2)$24, with the exceptional case $(v,v+2)$25 where $(v,v+2)$26 ties. This line of work situates Fibonacci-number terminology in extremal graph theory rather than in a single graph family (0802.3284, 0811.1449).

5. Additive, order-theoretic, and planar constructions

The Fibonacci-sum graph $(v,v+2)$27 has vertex set $(v,v+2)$28, with distinct vertices $(v,v+2)$29 adjacent if and only if $(v,v+2)$30 for some Fibonacci number $(v,v+2)$31. This family is bipartite for all $(v,v+2)$32, outerplanar for all $(v,v+2)$33, and has treewidth $(v,v+2)$34 for all $(v,v+2)$35. Its Hamiltonian-path behavior is rigid: $(v,v+2)$36 The automorphism group is also small: each $(v,v+2)$37 has at most one non-trivial automorphism, given explicitly in the classified cases (Arman et al., 2017).

The Young–Fibonacci graph is the Hasse diagram of the Young–Fibonacci lattice. Its vertices are words over $(v,v+2)$38, the rank of a vertex $(v,v+2)$39 is the digit sum $(v,v+2)$40, and the edges correspond to replacing the leftmost $(v,v+2)$41 by $(v,v+2)$42 or inserting a $(v,v+2)$43 to the left of the leftmost $(v,v+2)$44. For vertices $(v,v+2)$45, the number of downward paths $(v,v+2)$46 admits the explicit formula

$(v,v+2)$47

and the paper generalizes this to trajectories with prescribed upward steps. The resulting path-counting formula is polynomial with respect to the minimum of the ranks (Evtushevsky, 2020).

The $(v,v+2)$48-bonacci graphs arise from polyominoes associated with binary words avoiding $(v,v+2)$49 consecutive $(v,v+2)$50s. A word $(v,v+2)$51 determines a bargraph with $(v,v+2)$52 columns, where the $(v,v+2)$53-th column has $(v,v+2)$54 unit cells; the graph is formed by taking cell corners as vertices and cell sides as edges. These are chemical graphs, hence all vertices have degree at most four. The paper derives rational generating functions for the total number of vertices, edges, degree distributions, and Hamiltonian members of the family. It also stresses that these graphs are not the standard Fibonacci cubes: here the construction is polyomino-based rather than Hamming-adjacency-based (Kirgizov et al., 2022).

A different additive use of the term appears in the simple graph family with $(v,v+2)$55 vertices whose adjacency matrix is block diagonal with $(v,v+2)$56 blocks

$(v,v+2)$57

For this graph,

$(v,v+2)$58

so the number of walks of length $(v,v+2)$59 between the two vertices of a component is $(v,v+2)$60, and the number from a vertex to itself is $(v,v+2)$61. This is a direct walk-count model of Fibonacci recursion (Yılmaz et al., 2012).

6. Generalizations, corrections, and open directions

Recent work has emphasized both extension and correction. On the extension side, Munarini graphs provide a generalization of Fibonacci cubes and Pell graphs that preserves the daisy-cube property, and they satisfy

$(v,v+2)$62

through the standard daisy-cube relation between weight and cube polynomials. On the correction side, the distance cube polynomial generating function for Fibonacci-run graphs required revision because the earlier argument used $(v,v+2)$63 instead of the correct distance $(v,v+2)$64 for a $(v,v+2)$65 with top vertex $(v,v+2)$66 (Mollard, 14 May 2026, Mollard, 2024).

Open problems remain in several subareas. In the directed acyclic setting, the global optimality of the balanced decomposition method for Fibonacci-graph expressions is still posed as a conjecture. In the run-graph setting, the concluding discussion asks for a characterization of all hypercube subgraphs satisfying

$(v,v+2)$67

and of all graphs satisfying $(v,v+2)$68. The earlier basic study of Fibonacci-run graphs also conjectures that $(v,v+2)$69 always has a Hamiltonian path and that it is Hamiltonian exactly when $(v,v+2)$70 (Korenblit et al., 2013, Mollard, 2024, Eğecioğlu et al., 2020).

A persistent source of confusion is the assumption that “Fibonacci graph” names a single standard object. The literature here suggests instead a family of related but distinct paradigms: hypercube subgraphs defined by Fibonacci-type forbidden patterns, additive graphs with Fibonacci-determined adjacency, graded graphs whose ranks or path counts are Fibonacci-enumerated, and st-dags whose recursive source–sink structure is itself Fibonacci. The common content is not a unique graph but a recurring mechanism: constrained local structure giving rise to Fibonacci-number enumeration, Fibonacci-type generating functions, or Fibonacci-governed embeddings and invariants.