# Golden Laplacian Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**distance**between two vertices in G is the length (number of edges) of the shortest path connecting the two vertices. The

**diameter**D is the maximum of all distances between pairs of vertices in G.

**independent set**if no two of its vertices are adjacent. The largest cardinality of an independent set in G is the

**independence number**, $ind\left(G\right)$.

**Hamiltonian**if it contains a spanning cycle (Hamiltonian cycle). The graph is

**Hamiltonian connected**if any pair of vertices are the ends of a spanning path.

**pancyclic**if it contains cycles of all length l, $3\le l\le n$. Obviously, a pancyclic graph is Hamiltonian.

**vertex connectivity**of a connected graph ${\kappa}_{v}\left(G\right)$ is the minimum number of vertices for which their removal either disconnects G or reduces it to a single-vertex graph. A graph is

**k-connected**if ${\kappa}_{v}\left(G\right)\ge k$.

**clique number**$\omega \left(G\right)$ is the number of vertices in a largest clique of G.

**matching**in G is a set of mutually non-adjacent edges in G. A matching is

**perfect**if every vertex in G is incident to some edge in the matching. If the number of vertices of G is odd, the graph may contain a

**near-perfect matching**if exactly one vertex is unmatched.

**dominating set**of a graph G if each vertex in V is in S or is adjacent to a vertex in S. The

**domination number**$\gamma \left(G\right)$ is the minimum cardinality of a dominating set of G.

**isopermetric number**of a graph is defined as follows. Let $S\subset V$ and let $\partial S$ be the edge boundary of S, i.e., those edges with one endpoint inside S and another outside S. Then, the isopermetric number is

**Path graph**on n vertices, ${P}_{n}$: the graph with $n-2$ vertices of degree two and two vertices of degree one.

**Cycle graph**on n vertices, ${C}_{n}$: the graph with n vertices of degree two.

**Complete graph**on n vertices, ${K}_{n}$: the graph with all vertices of degree $n-1$.

**Complete graph minus an edge**, ${K}_{n}-e$: the complete graph ${K}_{n}$ on which an edge has been deleted.

**Complete bipartite graph**on n vertices, ${K}_{p,q}$: the graph on $n=p+q$ vertices which can be partitioned into two subsets of cardinalities p and q, respectively, such that no edge has both endpoints in the same subset and every possible edge that could connect vertices in different subsets is part of the graph.

**Complete split graph**on n vertices, ${S}_{n,\alpha}$: the graph on n vertices consisting of a clique on $n-\alpha $ vertices and an independent set on the remaining $\alpha $ ($1\le \alpha \le n-1$) vertices in which each vertex of the clique is adjacent to each vertex of the independent set.

**Lollipop graph**on n vertices, ${H}_{n,p}$: the graph on n vertices obtained by appending a cycle ${C}_{p}$ to a pendant vertex of a path ${P}_{n-p}$.

**Kite graph**on n vertices, ${Y}_{p,q}$: the graph on $n=p+q$ vertices obtained by appending a complete graph ${K}_{p}$ to a pendant vertex of a path ${P}_{q}$.

**Friendship graph**on n vertices, ${F}_{r}$: the graph on $n=2r+1$ vertices consisting of r triangles attached to a common vertex.

**Wheel graph**on n vertices, ${W}_{n}$: the graph on n vertices obtained by the graph join operation ${C}_{n-1}+{K}_{1}$, which consists of connecting every vertex of a cycle ${C}_{n-1}$ to a common vertex not in the cycle.

**Fan graph**on n vertices, ${A}_{n}$: the graph on n vertices resulting from the graph join operation ${P}_{n-1}+{K}_{1}$, consisting of joining every vertex of a path graph ${P}_{n-1}$ to a single vertex not in the cycle; we propose using the letter A from the Spanish “abanico”, meaning “fan”.

## 3. Golden Laplacian Spectra

**Definition**

**1.**

#### Properties of GLGs

**Lemma**

**1.**

- ${\mu}_{n}\left({G}_{n}\right)\le max\left\{{\displaystyle \frac{{k}_{u}\left({k}_{u}+{m}_{u}\right)+{k}_{v}\left({k}_{v}+{m}_{v}\right)}{{k}_{u}+{k}_{v}}},u,v\in E\right\}$, where ${m}_{u}$ is the average degree of the nearest neighbors of the vertex u (see [24]);
- ${\mu}_{2}\le {\displaystyle \frac{n-1}{n}}\delta $ (see [25]);
- ${\mu}_{2}\le {c}_{v}\le {c}_{e}\le \delta $, where ${c}_{v}$ and ${c}_{e}$ denote the vertex and edge connectivity, respectively (see [25]).

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Remark**

**1.**

**Lemma**

**10.**

**Proof.**

**Theorem**

**1.**

**Lemma**

**11.**

**Proof.**

## 4. Discovering GLGs

**Lemma**

**12.**

**Proof.**

**Lemma**

**13.**

**Proof.**

**Lemma**

**14.**

**Proof.**

**Lemma**

**15.**

**Proof.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Proof.**

#### Computer-Based Search

- There are no GLGs among the 112 connected graphs with $n=6$ vertices;
- There are no GLGs among the 853 connected graphs with $n=7$ vertices;
- There are 15 GLGs among the 11,117 connected graphs with $n=8$ vertices (see Figure 3).

- There are five GLGs among the 261,080 connected graphs with $n=9$ vertices, which are illustrated in Figure 4.

- There are 102 GLGs among the 11,716,571 connected graphs with $n=10$ vertices. (The adjacency matrices (in MATLAB format) and a table with the properties of GLGs with ten vertices can be requested to the main author via email).

- The fifteen GLG with eight vertices, five GLGs with nine vertices, and 102 GLGs with ten vertices have the following general properties:
- For all of these graphs, ${\mu}_{n}=5+\sqrt{5}$ and ${\mu}_{2}=5-\sqrt{5}$.
- They are all Hamiltonian; notice that all GLGs with $n\le 10$ are pancyclic except for the one with adjacency matrix $A\left({C}_{5}\right)\otimes {J}_{2}$, where ⊗ is the Kronecker product and ${J}_{2}$ is the all-ones matrix of order 2 (see the next section for details of this operation).
- They have a perfect (even n) or nearly-perfect matching (odd n), as evident from the fact that all of these GLGs have a Hamiltonian cycle, which implies the existence of a perfect matching.
- They have diameter 2.
- They have clique number $3\le \omega \le 4$.
- They have ${\kappa}_{v}={\kappa}_{e}=\delta $, except for the graph with adjacency matrix $\left(A\left({C}_{5}\right)+{I}_{5}\right)\otimes {J}_{2}-{I}_{10}$ (see the next section for details of this operation).
- They have minimum degree $\delta \ge 3$.
- They have domination number 2 ($n\le 9$) or $2\le \gamma \le 3$ ($n=10$); notice that if $\delta \ge 3$, then (according to Reed [32]) $\gamma \ge {\displaystyle \frac{3n}{8}}$, which is the bound observed for $n\le 10$.

## 5. Expanding the Family of GLGs

**Lemma**

**18.**

**Lemma**

**19.**

**Definition**

**2.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Proposition**

**1.**

**Theorem**

**4.**

_{r}is a GLG.

**Proof.**

## 6. Synchronization of GLGs

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 7. Conclusions and Future Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**–

**o**) Illustration of the 15 connected graphs with $n=8$ vertices which have golden Laplacian spectra.

**Figure 5.**Histogram of the values of the eigenratio $\mathcal{Q}$ of the 11,117 connected graphs with eight vertices. The graphs with Q equal to the square of the golden ratio are marked as a vertical broken line. Only those graphs having $\mathcal{Q}$ below this line are more synchronizable than GLGs.

**Figure 6.**Synchronization error E vs. coupling coefficient $\sigma $ for two networks of $n=10$ coupled Rossler oscillators, as in (80). Both GLGs (shown in the insets of panels (

**left**) and (

**right**)) display the same region of stability for synchronization, as $k=2$ for both of them. Red asterisks mark the predicted thresholds for synchronization based on the master stability function approach, as in Equation (79) with $k=2$.

# | m | $\mathit{\delta}$ | $\Delta $ | D | ind | ${\mathit{\kappa}}_{\mathit{v}}$ | ${\mathit{\kappa}}_{\mathit{e}}$ | $\mathit{\omega}$ | $\mathit{\gamma}$ | H | P | M | r | p |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | 16 | 4 | 4 | 2 | 3 | 4 | 4 | 3 | 2 | Y | Y | Y | Y | N |

b | 17 | 4 | 5 | 2 | 3 | 4 | 4 | 4 | 2 | Y | Y | Y | N | N |

c | 17 | 3 | 6 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

d | 18 | 3 | 6 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

e | 18 | 3 | 6 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

f | 18 | 3 | 5 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

g | 18 | 4 | 5 | 2 | 2 | 4 | 4 | 3 | 2 | Y | Y | Y | N | Y |

h | 18 | 4 | 5 | 2 | 2 | 4 | 4 | 4 | 2 | Y | Y | Y | N | N |

i | 19 | 3 | 6 | 2 | 2 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

j | 16 | 3 | 5 | 2 | 3 | 3 | 3 | 3 | 2 | Y | Y | Y | N | N |

k | 17 | 3 | 5 | 2 | 3 | 3 | 3 | 3 | 2 | Y | Y | Y | N | N |

l | 17 | 3 | 5 | 2 | 3 | 3 | 3 | 3 | 2 | Y | Y | Y | N | N |

m | 18 | 3 | 6 | 2 | 4 | 3 | 3 | 4 | 2 | Y | Y | Y | N | Y |

n | 18 | 3 | 6 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

o | 18 | 3 | 6 | 2 | 3 | 3 | 3 | 4 | 2 | Y | Y | Y | N | N |

# | m | $\mathit{\delta}$ | $\Delta $ | D | ind | ${\mathit{\kappa}}_{\mathit{v}}$ | ${\mathit{\kappa}}_{\mathit{e}}$ | $\mathit{\omega}$ | $\mathit{\gamma}$ | H | P | M | r | p |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | 19 | 4 | 5 | 2 | 3 | 4 | 4 | 3 | 2 | Y | Y | Y | N | N |

b | 21 | 4 | 5 | 2 | 3 | 4 | 4 | 4 | 2 | Y | Y | Y | N | N |

c | 20 | 4 | 5 | 2 | 3 | 4 | 4 | 3 | 2 | Y | Y | Y | N | N |

d | 20 | 4 | 5 | 2 | 3 | 4 | 4 | 4 | 2 | Y | Y | Y | N | N |

e | 19 | 4 | 5 | 2 | 3 | 4 | 4 | 3 | 2 | Y | Y | Y | N | N |

**Table 3.**List of a series of class III systems, along with their equations, coupling type, and parameters ${\nu}_{1}$, ${\nu}_{2}$, and ${\nu}_{2}/{\nu}_{1}$ for which ${\nu}_{2}/{\nu}_{1}<{\phi}^{2}$. All of these graphs are synchronizable when coupled by any GLG. For the Chua’s circuit, $f\left(x\right)=\left\{\begin{array}{cc}-bx-a+b,\hfill & x>1\\ -ax,\hfill & \left|x\right|<1\\ -bx-a+b,\hfill & x<-1\end{array}\right.$ with $a=-1.27$ and $b=-0.68$. The notation $i\to j$ indicates that the i-th variable of one oscillator is coupled to the dynamics of the j-th variable of the other oscillator (linear diffusive coupling is always assumed here). Data on the values of ${\nu}_{1}$ and ${\nu}_{2}$ are taken from [40].

System | Equations | Coupling | ${\mathit{\nu}}_{1}$ | ${\mathit{\nu}}_{2}$ | ${\mathit{\nu}}_{2}/{\mathit{\nu}}_{1}$ |
---|---|---|---|---|---|

Rossler |
$$\begin{array}{c}\dot{x}=-y-z\hfill \\ \dot{y}=x+0.2y\hfill \\ \dot{z}=0.2+(x-9)z\hfill \end{array}$$
| $1\to 1$ | 0.186 | 4.614 | 24.807 |

Lorenz |
$$\begin{array}{c}\dot{x}=10(y-x)\hfill \\ \dot{y}=x(28-z)-y\hfill \\ \dot{z}=xy-\frac{8}{3}z\hfill \end{array}$$
| $2\to 1$ | 3.98 | 23.692 | 5.953 |

Chen |
$$\begin{array}{c}\dot{x}=35(y-x)\hfill \\ \dot{y}=(-7-z)x+cy\hfill \\ \dot{z}=xy-\frac{8}{3}z\hfill \end{array}$$
| $3\to 3$ | 5.347 | 21.51 | 4.023 |

Chua |
$$\begin{array}{c}\dot{x}=10[y-x+f\left(x\right)]\hfill \\ \dot{y}=x-y+z\hfill \\ \dot{z}=-14.87y\hfill \end{array}$$
| $3\to 3$ | 0.788 | 4.82 | 6.117 |

Hindmarsh-Rose |
$$\begin{array}{c}\dot{x}=y+3{x}^{2}-{x}^{3}-z+3.2\hfill \\ \dot{y}=1-5{x}^{2}-y\hfill \\ \dot{z}=0.006[-z+4(x+1.6)]\hfill \end{array}$$
| $2\to 1$ | 0.286 | 1.233 | 4.311 |

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Akhter, S.; Frasca, M.; Estrada, E.
Golden Laplacian Graphs. *Mathematics* **2024**, *12*, 613.
https://doi.org/10.3390/math12040613

**AMA Style**

Akhter S, Frasca M, Estrada E.
Golden Laplacian Graphs. *Mathematics*. 2024; 12(4):613.
https://doi.org/10.3390/math12040613

**Chicago/Turabian Style**

Akhter, Sadia, Mattia Frasca, and Ernesto Estrada.
2024. "Golden Laplacian Graphs" *Mathematics* 12, no. 4: 613.
https://doi.org/10.3390/math12040613