Wreath product of graphs: topological indices and spectrum Alfredo - PowerPoint PPT Presentation

Motivations Preliminaries Results Wreath product of graphs: topological indices and spectrum Alfredo Donno Universit` a Niccol` o Cusano, Roma Workshop on Algebraic Graph Theory and Complex Networks Universit` a di Napoli Federico II - September, 13 2018

Motivations Preliminaries Results Motivations Preliminaries Results

Motivations Preliminaries Results GRAPH COMPOSITIONS � MATRIX COMPOSITIONS The correspondence is achieved by the notion of ADJACENCY MATRIX . Spectra of adjacency matrices and Laplacians are the main object of Spectral graph theory : connectivity, regularity and other graph invariants; expander graphs; random walks and rapidly mixing Markov chains; isospectrality problems; determination and characterization problem; applications to Mathematical Chemistry.

Motivations Preliminaries Results The adjacency matrix of a graph G = ( V G , E G ) undirected simple finite graph. The adjacency matrix of G is the matrix A G = ( a u , v ) u , v ∈ V G , � 1 if u ∼ v where a u , v = 0 if u �∼ v . Example   0 1 1 0 1 v 1  1 0 1 0 1  �� �� �� ��   G v 2 v 5 �� �� A G = 1 1 0 1 0   �� �� � � �� �� � �   0 0 1 0 1 �� �� � � v 3 �� �� 1 1 0 1 0 �� �� v 4 � �

Motivations Preliminaries Results The adjacency matrix of a graph G = ( V G , E G ) undirected simple finite graph. The adjacency matrix of G is the matrix A G = ( a u , v ) u , v ∈ V G , � 1 if u ∼ v where a u , v = 0 if u �∼ v . Example   0 1 1 0 1 v 1  1 0 1 0 1  �� �� �� ��   G v 2 v 5 �� �� A G = 1 1 0 1 0   �� �� � � �� �� � �   0 0 1 0 1 �� �� � � v 3 �� �� 1 1 0 1 0 �� �� v 4 � � Remarks: ♦ G undirected ⇒ A G symmetric; ♦ deg u = � v ∈ V G a u , v = number of vertices adjacent to u ; ♦ G d -regular ⇔ � v ∈ V G a u , v = d for each u ∈ V G .

Motivations Preliminaries Results Cartesian product of graphs Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two finite graphs. The Cartesian product G 1 � G 2 is the graph with: • vertex set V 1 × V 2 • where ( v 1 , v 2 ) ∼ ( w 1 , w 2 ) if: 1. either v 1 = w 1 and v 2 ∼ w 2 in G 2 ; 2. or v 2 = w 2 and v 1 ∼ w 1 in G 1 . Then: A G 1 � G 2 = I G 1 ⊗ A G 2 + A G 1 ⊗ I G 2 .

Motivations Preliminaries Results Example a � � � � � � G 1 G 2 � � � � �� �� � � �� �� 0 1 �� �� �� �� � � �� �� b c � � a , 0 a , 1 �� �� � �� �� c , 0 c , 1 G 1 � G 2 �� �� �� �� �� �� �� �� �� �� �� �� � b , 0 b , 1

Motivations Preliminaries Results Direct product of graphs Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two finite graphs. The direct product G 1 × G 2 is the graph with: • vertex set V 1 × V 2 • where ( v 1 , v 2 ) ∼ ( w 1 , w 2 ) if v 1 ∼ w 1 in G 1 and v 2 ∼ w 2 in G 2 . Then: A G 1 × G 2 = A G 1 ⊗ A G 2 . (Kronecker product of matrices)

Motivations Preliminaries Results Example a � � � � � � G 1 G 2 � � � � �� �� � � 0 �� �� 1 � � � � � � � � b c � � � � a , 0 c , 1 �� �� �� �� �� �� G 1 × G 2 b , 1 b , 0 �� �� � � � � � � � � � � � � c , 0 a , 1 � �

Motivations Preliminaries Results Strong product of graphs Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two finite graphs. The strong product G 1 ⊠ G 2 is the graph with: • vertex set V 1 × V 2 • where ( v 1 , v 2 ) ∼ ( w 1 , w 2 ) if: 1. v 1 = w 1 and v 2 ∼ w 2 in G 2 ; 2. or v 2 = w 2 and v 1 ∼ w 1 in G 1 ; 3. or v 1 ∼ w 1 in G 1 and v 2 ∼ w 2 in G 2 . Then: A G 1 ⊠ G 2 = I G 1 ⊗ A G 2 + A G 1 ⊗ I G 2 + A G 1 ⊗ A G 2 .

Motivations Preliminaries Results Lexicographic product of graphs Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be two finite graphs. The lexicographic product G 1 ◦ G 2 is the graph with: • vertex set V 1 × V 2 • where ( v 1 , v 2 ) ∼ ( w 1 , w 2 ) if: 1. either v 1 ∼ w 1 in G 1 ; 2. or v 1 = w 1 and v 2 ∼ w 2 in G 2 . Then: A G 1 ◦ G 2 = A G 1 ⊗ J G 2 + I G 1 ⊗ A G 2 , where J G 2 is the matrix indexed by V 2 whose entries are all equal to 1.