Are almost all graphs determined by their spectrum? Aida Abiad - - PowerPoint PPT Presentation

Are almost all graphs determined by their spectrum?

Aida Abiad Tilburg University, The Netherlands

Introduction The problem

A church, a graph, and its adjacency matrix

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

A church, a graph, and its adjacency matrix spectrum: 3, 1, 1, 1, 1, 1, −2, −2, −2, −2

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

A (finite simple) graph G on n vertices ⇓ The spectrum λ1 ≥ · · · ≥ λn of the adjacency matrix A of G

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

A (finite simple) graph G on n vertices ⇑ ? The spectrum λ1 ≥ · · · ≥ λn of the adjacency matrix A of G

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

The following graph properties can be deduced from the spectrum: number of nodes (= n) number of edges (= 1

2

• i λ2

i )

number of triangles (= 1

6

• i λ3

i )

bipartite (⇔ λi = −λn−i+1, i = 1, . . . , n) regular (⇔ λ1 = 1

n

• i λ2

i )

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Cospectral graphs

      1 1 1 1 1 1 1 1             1 1 1 1 1 1 1 1      

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Cospectral graphs

      1 1 1 1 1 1 1 1             1 1 1 1 1 1 1 1      

spectrum: −2, 0, 0, 0, 2

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

A graph G is said to be determined by its spectrum (DS) if every graph cospectral with G is isomorphic with G.

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Examples of DS graphs:

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Examples of DS graphs:

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Examples of DS graphs:

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Conjecture [van Dam and Haemers, 2003] Almost all graphs are determined by their spectrum (DS).

Aida Abiad Are almost all graphs determined by their spectrum?

Introduction The problem

Thanks for your attention

Aida Abiad Are almost all graphs determined by their spectrum?