PROPERTIES OF SOME ALGEBRAICALLY DEFINED DIGRAPHS Aleksandr Kodess, - PowerPoint PPT Presentation

PROPERTIES OF SOME ALGEBRAICALLY DEFINED DIGRAPHS Aleksandr Kodess, Felix Lazebnik Department of Mathematical Sciences University of Delaware Modern Trends of Algebraic Graph Theory Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is a digraph? Definition v 1 v 5 A digraph is a pair D = ( V , A ) of: a set V , whose elements are called vertices or nodes v 2 v 4 a set A of ordered pairs of vertices, called arcs , directed edges , or arrows v 3 Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is an algebraic digraph D ( q ; f ) ? Let F q be a finite field with q elements; f : F 2 q → F q be a bivariate polynomial. Definition An algebraic digraph , denoted D ( q ; f ) , is a digraph whose vertex set is F 2 q � � arc set consits of ordered pairs ( x 1 , x 2 ) , ( y 1 , y 2 ) with the relation x 2 + y 2 = f ( x 1 , y 1 ) , Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is an algebraic digraph D ( q ; f ) ? Let F q be a finite field with q elements; f : F 2 q → F q be a bivariate polynomial. Definition An algebraic digraph , denoted D ( q ; f ) , is a digraph whose vertex set is F 2 q � � arc set consits of ordered pairs ( x 1 , x 2 ) , ( y 1 , y 2 ) with the relation x 2 + y 2 = f ( x 1 , y 1 ) , Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is an algebraic digraph D ( q ; f ) ? Let F q be a finite field with q elements; f : F 2 q → F q be a bivariate polynomial. Definition An algebraic digraph , denoted D ( q ; f ) , is a digraph whose vertex set is F 2 q � � arc set consits of ordered pairs ( x 1 , x 2 ) , ( y 1 , y 2 ) with the relation x 2 + y 2 = f ( x 1 , y 1 ) , Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is an algebraic digraph D ( q ; f ) ? Let F q be a finite field with q elements; f : F 2 q → F q be a bivariate polynomial. Definition An algebraic digraph , denoted D ( q ; f ) , is a digraph whose vertex set is F 2 q � � arc set consits of ordered pairs ( x 1 , x 2 ) , ( y 1 , y 2 ) with the relation x 2 + y 2 = f ( x 1 , y 1 ) , Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is an algebraic digraph D ( q ; f ) ? Let F q be a finite field with q elements; f : F 2 q → F q be a bivariate polynomial. Definition An algebraic digraph , denoted D ( q ; f ) , is a digraph whose vertex set is F 2 q � � arc set consits of ordered pairs ( x 1 , x 2 ) , ( y 1 , y 2 ) with the relation x 2 + y 2 = f ( x 1 , y 1 ) , Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Example of D ( q ; f ) Example of D ( q ; f ) V ( D ) = F 2 q f : F 2 q → F q There is an arc from vertex ( x 1 , x 2 ) to vertex ( y 1 , y 2 ) if and only if x 2 + xy + y 2 + 1 x 2 + y 2 = = f 1 ( x 1 , y 1 ) x 2 + xy + y 2 x 2 + y 2 = = f 2 ( x 1 , y 1 ) x 2 + y 2 = xy = f 3 ( x 1 , y 1 ) Easy to argue that if q is odd, then D ( q ; f 1 ) ∼ = D ( q ; f 2 ) ∼ = D ( q ; f 3 ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Example of D ( q ; f ) Example of D ( q ; f ) V ( D ) = F 2 q f : F 2 q → F q There is an arc from vertex ( x 1 , x 2 ) to vertex ( y 1 , y 2 ) if and only if x 2 + xy + y 2 + 1 x 2 + y 2 = = f 1 ( x 1 , y 1 ) x 2 + xy + y 2 x 2 + y 2 = = f 2 ( x 1 , y 1 ) x 2 + y 2 = xy = f 3 ( x 1 , y 1 ) Easy to argue that if q is odd, then D ( q ; f 1 ) ∼ = D ( q ; f 2 ) ∼ = D ( q ; f 3 ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Example of D ( q ; f ) Example of D ( q ; f ) V ( D ) = F 2 q f : F 2 q → F q There is an arc from vertex ( x 1 , x 2 ) to vertex ( y 1 , y 2 ) if and only if x 2 + xy + y 2 + 1 x 2 + y 2 = = f 1 ( x 1 , y 1 ) x 2 + xy + y 2 x 2 + y 2 = = f 2 ( x 1 , y 1 ) x 2 + y 2 = xy = f 3 ( x 1 , y 1 ) Easy to argue that if q is odd, then D ( q ; f 1 ) ∼ = D ( q ; f 2 ) ∼ = D ( q ; f 3 ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Example of D ( q ; f ) Example of D ( q ; f ) V ( D ) = F 2 q f : F 2 q → F q There is an arc from vertex ( x 1 , x 2 ) to vertex ( y 1 , y 2 ) if and only if x 2 + xy + y 2 + 1 x 2 + y 2 = = f 1 ( x 1 , y 1 ) x 2 + xy + y 2 x 2 + y 2 = = f 2 ( x 1 , y 1 ) x 2 + y 2 = xy = f 3 ( x 1 , y 1 ) Easy to argue that if q is odd, then D ( q ; f 1 ) ∼ = D ( q ; f 2 ) ∼ = D ( q ; f 3 ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Simple Observation Simple isomorphisms Let q be an odd prime power, and f ∈ F q [ x , y ] . Let f 1 ( x , y ) = f ( x , y ) − f ( 0 , 0 ) , and f ∗ ( x , y ) = f 1 ( x , y ) − f ( x , 0 ) − f ( 0 , y ) . The following statements hold: D ( q ; f ) ∼ = D ( q ; f 1 ) . If, in addition, f is a symmetric polynomial, then D ( q ; f ) ∼ = D ( q ; f 1 ) ∼ = D ( q ; f ∗ ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Simple Observation Simple isomorphisms Let q be an odd prime power, and f ∈ F q [ x , y ] . Let f 1 ( x , y ) = f ( x , y ) − f ( 0 , 0 ) , and f ∗ ( x , y ) = f 1 ( x , y ) − f ( x , 0 ) − f ( 0 , y ) . The following statements hold: D ( q ; f ) ∼ = D ( q ; f 1 ) . If, in addition, f is a symmetric polynomial, then D ( q ; f ) ∼ = D ( q ; f 1 ) ∼ = D ( q ; f ∗ ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Simple Observation Simple isomorphisms Let q be an odd prime power, and f ∈ F q [ x , y ] . Let f 1 ( x , y ) = f ( x , y ) − f ( 0 , 0 ) , and f ∗ ( x , y ) = f 1 ( x , y ) − f ( x , 0 ) − f ( 0 , y ) . The following statements hold: D ( q ; f ) ∼ = D ( q ; f 1 ) . If, in addition, f is a symmetric polynomial, then D ( q ; f ) ∼ = D ( q ; f 1 ) ∼ = D ( q ; f ∗ ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is a monomial algebraic digraph? Definition A monomial algebraic digraph , denoted D ( q ; m , n ) , is an algebraic digraph in which vertex set V is F 2 q there is an arc from ( x 1 , x 2 ) to ( y 1 , y 2 ) if and only if x 2 + y 2 = x m 1 y n 1 . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

D ( 3 ; 1 , 2 ) � 2,1 � � 0,2 � � 1,1 � � 1,0 � � 0,0 � � 2,2 � � 2,0 � � 0,1 � � 1,2 � Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Motivation Work of: Lazebnik, Woldar (2001) Lazebnik, Ustimenko (1993, 1995, 1996) Viglione (2001) Dmytrenko, Lazebnik, Viglione (2005) Bipartite undirected graph B Γ n V ( B Γ n ) = P n ∪ L n , both P n and L n are copies of F n q point ( p ) = ( p 1 , . . . , p n ) is adjacent to line [ l ] = ( l 1 , . . . , l n ) if l 2 + p 2 = f 2 ( p 1 , l 1 ) l 3 + p 3 = f 3 ( p 1 , l 1 , p 2 , l 2 ) . . . l n + p n = f n ( p 1 , l 1 , p 2 , l 2 , . . . , p n − 1 , l n − 1 ) . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Properties of B Γ n B Γ n admits neighbor-complete coloring , i.e. every color is uniquely represented among the neighbors of each vertex covering properties of B Γ n . For instance, B Γ n covers B Γ k for n > k . embedded spectra properties. For instance, spec ( B Γ k ) ⊆ spec ( B Γ n ) for k < n . edge-decompostiion properties. We have B Γ n decomposing K q n , q n . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Properties of B Γ n B Γ n admits neighbor-complete coloring , i.e. every color is uniquely represented among the neighbors of each vertex covering properties of B Γ n . For instance, B Γ n covers B Γ k for n > k . embedded spectra properties. For instance, spec ( B Γ k ) ⊆ spec ( B Γ n ) for k < n . edge-decompostiion properties. We have B Γ n decomposing K q n , q n . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Properties of B Γ n B Γ n admits neighbor-complete coloring , i.e. every color is uniquely represented among the neighbors of each vertex covering properties of B Γ n . For instance, B Γ n covers B Γ k for n > k . embedded spectra properties. For instance, spec ( B Γ k ) ⊆ spec ( B Γ n ) for k < n . edge-decompostiion properties. We have B Γ n decomposing K q n , q n . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

Properties of B Γ n B Γ n admits neighbor-complete coloring , i.e. every color is uniquely represented among the neighbors of each vertex covering properties of B Γ n . For instance, B Γ n covers B Γ k for n > k . embedded spectra properties. For instance, spec ( B Γ k ) ⊆ spec ( B Γ n ) for k < n . edge-decompostiion properties. We have B Γ n decomposing K q n , q n . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

An application of B Γ n with a certain specialization Let C n denote the cycle of length n ≥ 3 ex ( v , { C 3 , C 4 , . . . , C 2 k } ) denote the greatest number of edges in a graph or order v which contains no subgraphs isomorphic to any C 3 , . . . , C 2 k . Theorem (Lazebnik, Ustimenko, Woldar 1995) 2 ex ( v , { C 3 , C 4 , . . . , C 2 k } ) ≥ c k v 1 + 3 k − 3 + ǫ , where c k is a positive function if k, and ǫ = 0 if k � = 5 is odd, and ǫ = 1 if k is even. This lower bounds comes from B Γ n with a certain choice of defining functions f i , 2 ≤ i ≤ n . Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs