Switching Checkerboards

Switching Checkerboards

David Ellison1 Bertrand Jouve2 Lewi Stone1

1_{RMIT University, School of Science} 2_{Universit´e de Toulouse, CNRS}

1 Introduction

Zagreb Indices

Description of the Problem

2 Switching a Checkerboard

Checkerboards Graph of Matrices Positive Checkerboards

3 Successive Positive Switches

Simulations and Observations Overall variation of λ1

Zagreb Indices

Let G = (V , E ) be a simple graph with n vertices and m edges.

M1= P i ∈V d_i2 M2= P {ij}∈E didj, Z1 = q M1 n ≤ λ1 Z2 = q M2 m

Z1 is the quadratic average of the degrees and a lower bound

of the spectral radius λ1.

Z2 is an average over the edges ofpdidj and gives a loose

Description of the problem

We consider simple undirected graphs. We fix a degree distribution.

Checkerboard

A checkerboard in a graph is an alternating cycle of 2 edges and 2 non-edges. The operation of switching a checkerboard consists of switching the edges and non-edges.

In the adjacency matrix, the switch affects a 2 × 2 sub-matrix and its symmetric: 1 0 0 1  ←→ 0 1 1 0 

Graph of matrices

Theorem (Ryser, 1963)

Any two graphs with identical degree distribution can be linked through sequence of switches.

Orienting the Graph of matrices

By distinguishing positive checkerboards1 0 0 1  and negative checkerboards0 1 1 0  , and only allowing switches from negative to positive, we define an orientation for the graph of matrices.

Proposition

The directed graph of matrices is acyclic.

Idea of proof: The quantity I (A) =P ij a_ij increases along the

Effect of a switch

Let A0 be obtained from A by switching a checkerboard of

coordinates (i , j , k, l ). Let X and X0 be the normalised principal eigenvectors of A and A0, respectively.

We have: λ0₁− λ1 = tX0A0X0− tXAX ≥ t_XA0_{X −}t_{XAX =} t_{X (A}0_{−A)X .} Thus λ0₁− λ1≥ 2(xi− xj)(xk− xl). 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 xi xk xj xl xi xk xj xl Similarly, M₂0 − M2 = (di− dj)(dk− dl). Remark

If the rows and columns of A are ordered increasingly (or

decreasingly), M2 increases with each positive switch and λ1

Sources and Sinks

Theorem

The sources and sinks of the directed graph of matrices are,

respectively, the local minima and maxima for M2 and Z2.

Theorem

The global maximum of λ1 is reached at a sink of the directed

graph of matrices.

Key to the proof: If λ1 is maximised, the coefficients of the

Simulations

Starting from a random graph with 20 vertices and 25 edges, successive random positive switches are applied.

Simulations

Starting from an Erd˝os-R´enyi graph with 50 vertices (p = .2),

Observations

Z2 is non-decreasing, but λ2 may locally decrease slightly. (As

expected)

Generally, for large graphs, the amplitude of the total variation is fairly small. However, some theoretical examples yield arbitrarily large variations.

For regular graphs, λ1 and Z2 do not vary at all.

Large variations of λ

1 λ1 = 2

−→

λ1= 4 Degree distribution {4,4,4,4,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} nK1,n−1 −→ Kn+ n(n−1)₂ K2 λ1= Z2 = √ n − 1 << λ1= n − 1, Z2≈ n−1√₂

Concavity

The observed concavity of λ1 may be explained by the following

result:

Theorem

For an Erd˝os-R´enyi graph, the average variation of λ1 caused by a

positive switch decreases on average after a positive switch.

Elements of proof: The average variation of λ caused by a switch is twice the average checkerboard area (xi − xj)(xk − xl). After a

positive switch, some remaining checkerboards have their area reduced, but some small checkerboards disappear. The key is to verify that the former has a greater effect than the latter on the average checkerboard area.

Note that some carefully crafted pathological examples yield a convex increase of λ1.

Discrepancy between Z

2

and λ

1

For kite graphs, λ1 can be arbitrarily greater than Z2. (Stevanovi´c & Hansen, 2008)

For double stars, Z2

λ1 can be arbitrarily close to

q

3

2. (Abdo et al., 2014)

For certain cographs also, Z2

λ1 can be arbitrarily close to

q

3 2.

Conjecture: Revisiting a problem by Nikiforov (2006)

For any graph, Z2

λ1 <

q

3

2. Furthermore, for a fixed n, the

maximum of Z2

Questions