Friendship in Edge-Regular Graphs Kelly Bragan; James Hammer; Pete - PowerPoint PPT Presentation

Outline Introduction Description of all ER ( n ❀ 4 ❀ 1 ) Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Friendship in Edge-Regular Graphs Kelly Bragan; James Hammer; Pete Johnson, Auburn University; and Ken Roblee, Troy University February 27, 2011 James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Introduction Description of all ER ( n ❀ 4 ❀ 1 ) Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Table of Contents Introduction Background Preliminary Results General Form Description of all ER ( n ❀ 4 ❀ 1 ) Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) James M. Hammer, III Friendship in Edge-Regular Graphs

✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Basic Concepts Definition A simple graph G is Edge-Regular with parameters ( n ❀ d ❀ ✕ ) (We say G ✷ ER ( n ❀ d ❀ ✕ ) ) if and only if James M. Hammer, III Friendship in Edge-Regular Graphs

✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Basic Concepts Definition A simple graph G is Edge-Regular with parameters ( n ❀ d ❀ ✕ ) (We say G ✷ ER ( n ❀ d ❀ ✕ ) ) if and only if ◮ G has n vertices James M. Hammer, III Friendship in Edge-Regular Graphs

✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Basic Concepts Definition A simple graph G is Edge-Regular with parameters ( n ❀ d ❀ ✕ ) (We say G ✷ ER ( n ❀ d ❀ ✕ ) ) if and only if ◮ G has n vertices ◮ G is regular of degree d James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Basic Concepts Definition A simple graph G is Edge-Regular with parameters ( n ❀ d ❀ ✕ ) (We say G ✷ ER ( n ❀ d ❀ ✕ ) ) if and only if ◮ G has n vertices ◮ G is regular of degree d ◮ Any two adjacent vertices have exactly ✕ common neighbors James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Examples Figure: Degree 4, ✕ = 1 James M. Hammer, III Friendship in Edge-Regular Graphs

❀ ❀ ❀ ✕ ✻ ❀ ✕ ❃ ✕ � ✕ ✕ ❃ � ✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 1 of 2) Remark ◮ The graph on the left is the line graph of K 3 ❀ 3 , denoted L ( K 3 ❀ 3 ) . James M. Hammer, III Friendship in Edge-Regular Graphs

❀ ❀ ✕ ✻ ❀ ✕ ❃ ✕ � ✕ ✕ ❃ � ✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 1 of 2) Remark ◮ The graph on the left is the line graph of K 3 ❀ 3 , denoted L ( K 3 ❀ 3 ) . ◮ L ( K 3 ❀ 3 ) can also be viewed as K 3 � K 3 James M. Hammer, III Friendship in Edge-Regular Graphs

✕ ❃ � ✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 1 of 2) Remark ◮ The graph on the left is the line graph of K 3 ❀ 3 , denoted L ( K 3 ❀ 3 ) . ◮ L ( K 3 ❀ 3 ) can also be viewed as K 3 � K 3 ◮ If ER ( n ❀ d ❀ ✕ ) ✻ = ❀ and ✕ ❃ 0 , then n ✕ 3 ( d � ✕ ) , (Johnson, Roblee) James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 1 of 2) Remark ◮ The graph on the left is the line graph of K 3 ❀ 3 , denoted L ( K 3 ❀ 3 ) . ◮ L ( K 3 ❀ 3 ) can also be viewed as K 3 � K 3 ◮ If ER ( n ❀ d ❀ ✕ ) ✻ = ❀ and ✕ ❃ 0 , then n ✕ 3 ( d � ✕ ) , (Johnson, Roblee) ◮ The graph on the left represents an edge-regular graphs with ✕ ❃ 0 such that n = 3 ( d � ✕ ) . (Johnson, Roblee, and Smotzer) James M. Hammer, III Friendship in Edge-Regular Graphs

✕ ✏ ✑ ✕ ❀ ✕ ❀ ✕ ✷ ✕ ❀ ✕ ✕ ✕ ✕ ❀ ✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 2 of 2) Remark ◮ L ( K 3 ❀ 3 ) is the only graph in ER ( 9 ❀ 4 ❀ 1 ) , satisfying the above with equality. James M. Hammer, III Friendship in Edge-Regular Graphs

✕ ✕ ✕ ❀ ✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 2 of 2) Remark ◮ L ( K 3 ❀ 3 ) is the only graph in ER ( 9 ❀ 4 ❀ 1 ) , satisfying the above with equality. ◮ There exists an edge-regular graph for every integer value of ✕ . ( ✕ + 2 ) 2 ❀ 2 ✕ + 2 ❀ ✕ ✏ ✑ Take L ( K ( ✕ + 2 ) ❀ ( ✕ + 2 ) ) ✷ ER . James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Observations (Part 2 of 2) Remark ◮ L ( K 3 ❀ 3 ) is the only graph in ER ( 9 ❀ 4 ❀ 1 ) , satisfying the above with equality. ◮ There exists an edge-regular graph for every integer value of ✕ . ( ✕ + 2 ) 2 ❀ 2 ✕ + 2 ❀ ✕ ✏ ✑ Take L ( K ( ✕ + 2 ) ❀ ( ✕ + 2 ) ) ✷ ER . ◮ L ( K ( ✕ + 2 ) ❀ ( ✕ + 2 ) ) can be viewed as K ✕ + 2 � K ✕ + 2 James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Expanding ER ( n ❀ 4 ❀ 1 ) Figure: Expanding Degree 4 ◮ The graph on the right indicates that there is a graph in ER ( 3 + 9 s ❀ 4 ❀ 1 ) ❀ for each s ✷ N James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Expanding ER ( n ❀ 4 ❀ 1 ) Figure: Expanding Degree 4 ◮ The graph on the right indicates that there is a graph in ER ( 3 + 9 s ❀ 4 ❀ 1 ) ❀ for each s ✷ N ◮ Strip away the outer K 3 of the left graph and treat that as you did the original inside K 3 (in red.) James M. Hammer, III Friendship in Edge-Regular Graphs

❴ ✿ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Preliminary Results Remark ◮ It is easy to see that the closed neighborhood of each vertex v must be a friendship graph, K 1 ❴ d 2 K 2 ✿ James M. Hammer, III Friendship in Edge-Regular Graphs

❴ ✿ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Preliminary Results Remark ◮ It is easy to see that the closed neighborhood of each vertex v must be a friendship graph, K 1 ❴ d 2 K 2 ✿ ◮ It follows from this observation that the degree of each vertex is even. James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Preliminary Results Remark ◮ It is easy to see that the closed neighborhood of each vertex v must be a friendship graph, K 1 ❴ d 2 K 2 ✿ ◮ It follows from this observation that the degree of each vertex is even. Figure: K 1 ❴ d 2 K 2 ✿ James M. Hammer, III Friendship in Edge-Regular Graphs

✕ Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Starter Remark It is necessarily the case then that the following diagram is forced for d = 4 . This can be generalized to higher degrees by adding K 3 ’s at each vertex of the base K 3 . James M. Hammer, III Friendship in Edge-Regular Graphs

Outline Background Introduction Preliminary Results Description of all ER ( n ❀ 4 ❀ 1 ) General Form Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Starter Remark It is necessarily the case then that the following diagram is forced for d = 4 . This can be generalized to higher degrees by adding K 3 ’s at each vertex of the base K 3 . Figure: Degree 4, ✕ = 1 Starter James M. Hammer, III Friendship in Edge-Regular Graphs

✷ ❀ ❀ ✘ Outline Introduction Description of all ER ( n ❀ 4 ❀ 1 ) Preliminary work on ER ( n ❀ d ❀ ✕ ❃ 2 ) Description Theorem G ✷ ER ( n ❀ 4 ❀ 1 ) for some n ✷ N if and only if G is the line graph of a cubic triangle-free graph. James M. Hammer, III Friendship in Edge-Regular Graphs