# On the critical group of the missing Moore graph

@article{Ducey2017OnTC, title={On the critical group of the missing Moore graph}, author={Joshua E. Ducey}, journal={Discret. Math.}, year={2017}, volume={340}, pages={1104-1109} }

We consider the critical group of a hypothetical Moore graph of diameter $2$ and valency $57$. Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow $p$-subgroups of the critical group must be elementary abelian with the exception of $p = 5$. We prove that the $5$-rank of the Laplacian matrix determines the critical group up to two possibilities.

#### Topics from this paper

#### 7 Citations

Critical group structure from the parameters of a strongly regular graph

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2021

Many examples are given, including how the theory can be used to compute the critical group of Conway's $99$-graph and to give an elementary argument that no $srg(28,9,0,4)$ exists. Expand

The critical group of the Kneser graph on 2-subsets of an n-element set

- Mathematics
- Linear Algebra and its Applications
- 2018

In this paper we compute the critical group of the Kneser graph $KG(n,2)$. This is equivalent to computing the Smith normal form of a Laplacian matrix of this graph.

The sandpile group of polygon rings and twisted polygon rings

- Mathematics
- 2020

Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching… Expand

The sandpile group of a polygon flower

- Mathematics, Computer Science
- Discret. Appl. Math.
- 2019

The graph structure is used to find a set of generators and a relation matrix called R, which has the same form for any $F$ and has much smaller size than that of the (reduced) Laplacian matrix. Expand

Some constraints on the missing Moore graph

- Computer Science
- Australas. J Comb.
- 2020

Some constraints on the structure of the missing Moore graph are derived and it is shown that the graph can be partitioned into 2D and 3D models with respect to discrete-time components. Expand

MooreCube: A Scalable and Flexible Architecture for Cloud Computing Data Centers on Multi-port Servers

- Computer Science
- Scalable Comput. Pract. Exp.
- 2017

A new architecture called MooreCube is proposed that can increase network scalability and decrease network diameter as well as increase flexibility, and is a recursively defined architecture that uses Moore graph as Building Block structure and uses the hierarchical structure to meet high scalability. Expand

Scalable Computing: Practice and Experience

- 2016

The plethora of sensors deployed in Internet of Things (IoT) environments generate unprecedented volumes of data, thereby creating a data deluge. Data collected from these sensors can be used to… Expand

#### References

SHOWING 1-10 OF 17 REFERENCES

Search for properties of the missing Moore graph

- Mathematics
- 2010

In the degree-diameter problem, the only extremal graph the existence of which is still in doubt is the Moore graph of order 3250, degree 57 and diameter 2. It has been known that such a graph cannot… Expand

On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs

- Mathematics
- 1992

AbstractLet Γ be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over … Expand

On Moore Graphs with Diameters 2 and 3

- Mathematics, Computer Science
- IBM J. Res. Dev.
- 1960

The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph to prove the existence of connected, undirected graphs homogeneous of degree d and of diameter k. Expand

A finite group attached to the Laplacian of a graph

- Computer Science, Mathematics
- Discret. Math.
- 1990

It is shown that h is bounded by the number of independent cycles of G and some cases where these two integers are equal. Expand

On the pRank of the Adjacency Matrices of Strongly Regular Graphs

- 2003

Let F be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over Fp, the… Expand

Chip-Firing and the Critical Group of a Graph

- Mathematics
- 1999

A variant of the chip-firing game on a graph is defined. It is shown that the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and… Expand

Algebraic Graph Theory

- Mathematics, Computer Science
- Graduate texts in mathematics
- 2001

The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial. Expand

A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

- Mathematics, Computer Science
- J. Graph Theory
- 2016

It is shown that the method developed is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and it is argued that it can be used in many other cases when dealing with graphs that have numerous structural restrictions. Expand

Spectra of Graphs

- Computer Science
- 2011

This book gives an elementary treatment of the basic material about graph Spectra, both for ordinary, and Laplace and Seidel spectra, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Expand

Abstract Algebra

- Nature
- 1967

Abstract AlgebraBy Andrew O. Lindstrum jun. (Holden-Day Series in Mathematics.) Pp. xii + 211. (San Francisco and London: Holden-Day, Inc., 1967.) $10.