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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

Examples

Figure: Degree 4, ✕ = 1

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) Background Preliminary Results General Form

Observations (Part 1 of 2)

Remark

◮ The graph on the left is the line graph of K3❀3, denoted L(K3❀3).

❀

❀ ❀ ✕ ✻ ❀ ✕ ❃ ✕ ✕ ✕ ❃ ✕

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) Background Preliminary Results General Form

Observations (Part 1 of 2)

Remark

◮ The graph on the left is the line graph of K3❀3, denoted L(K3❀3). ◮ L(K3❀3) can also be viewed as K3K3

❀ ❀ ✕ ✻ ❀ ✕ ❃ ✕ ✕ ✕ ❃ ✕

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) Background Preliminary Results General Form

Observations (Part 1 of 2)

Remark

◮ The graph on the left is the line graph of K3❀3, denoted L(K3❀3). ◮ L(K3❀3) can also be viewed as K3K3 ◮ If ER(n❀ d❀ ✕) ✻= ❀ and ✕ ❃ 0, then n ✕ 3(d ✕), (Johnson, Roblee)

✕ ❃ ✕

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) Background Preliminary Results General Form

Observations (Part 1 of 2)

Remark

◮ The graph on the left is the line graph of K3❀3, denoted L(K3❀3). ◮ L(K3❀3) can also be viewed as K3K3 ◮ 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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

Observations (Part 2 of 2)

Remark

◮ L(K3❀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 Introduction Description of all ER(n❀ 4❀ 1) Preliminary work on ER(n❀ d❀ ✕ ❃ 2) Background Preliminary Results General Form

Observations (Part 2 of 2)

Remark

◮ L(K3❀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 ✕.

Take L(K(✕+2)❀(✕+2)) ✷ ER

✏

(✕ + 2)2 ❀ 2✕ + 2❀ ✕

✑

.

✕ ❀ ✕ ✕ ✕

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) Background Preliminary Results General Form

Observations (Part 2 of 2)

Remark

◮ L(K3❀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 ✕.

Take L(K(✕+2)❀(✕+2)) ✷ ER

✏

(✕ + 2)2 ❀ 2✕ + 2❀ ✕

✑

.

◮ L(K(✕+2)❀(✕+2)) can be viewed as K✕+2K✕+2

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) Background Preliminary Results General Form

Expanding ER(n❀ 4❀ 1)

Figure: Expanding Degree 4

◮ The graph on the right indicates that there is a graph in

ER(3 + 9s❀ 4❀ 1)❀ for each s ✷ N

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) Background Preliminary Results General Form

Expanding ER(n❀ 4❀ 1)

Figure: Expanding Degree 4

◮ The graph on the right indicates that there is a graph in

ER(3 + 9s❀ 4❀ 1)❀ for each s ✷ N

◮ Strip away the outer K3 of the left graph and treat that as you

did the original inside K3 (in red.)

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) Background Preliminary Results General Form

Preliminary Results

Remark

◮ It is easy to see that the closed neighborhood of each vertex v

must be a friendship graph, K1 ❴ d

2 K2✿

❴ ✿

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) Background Preliminary Results General Form

Preliminary Results

Remark

◮ It is easy to see that the closed neighborhood of each vertex v

must be a friendship graph, K1 ❴ d

2 K2✿ ◮ It follows from this observation that the degree of each vertex

is even.

❴ ✿

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) Background Preliminary Results General Form

Preliminary Results

Remark

◮ It is easy to see that the closed neighborhood of each vertex v

must be a friendship graph, K1 ❴ d

2 K2✿ ◮ It follows from this observation that the degree of each vertex

is even.

Figure: K1 ❴ d

2 K2✿

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) Background Preliminary Results General Form

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 K3’s at each vertex of the base K3.

✕

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) Background Preliminary Results General Form

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 K3’s at each vertex of the base K3.

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

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.

Proof.

◮ Sufficiency is easy to see.

✷ ❀ ❀ ✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

✷ ❀ ❀ ✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

◮ Let G ✷ ER(n❀ 4❀ 1).

✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

◮ Let G ✷ ER(n❀ 4❀ 1). ◮ Form H such that:

✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

◮ Let G ✷ ER(n❀ 4❀ 1). ◮ Form H such that: ◮ Vertices in H correspond to triangles in G

✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

◮ Let G ✷ ER(n❀ 4❀ 1). ◮ Form H such that: ◮ Vertices in H correspond to triangles in G ◮ Two vertices in H being adjacent if and only if the triangles

have a vertex in common in G.

✘

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.

Proof.

◮ Sufficiency is easy to see. ◮ Necessity:

◮ Let G ✷ ER(n❀ 4❀ 1). ◮ Form H such that: ◮ Vertices in H correspond to triangles in G ◮ Two vertices in H being adjacent if and only if the triangles

have a vertex in common in G.

◮ This graph is definitely cubic, triangle-free, and it can be

shown that G ✘ = L(H).

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)

Example

Figure: d = 4 pre-line graph

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)

Observations

Remark

For ✕ = 2, the graph on the left is the line graph of K4❀4. The graph on the right, however, does not contain any K4’s.

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)

Observations

Remark

For ✕ = 2, the graph on the left is the line graph of K4❀4. The graph on the right, however, does not contain any K4’s.

Figure: Degree 6

James M. Hammer, III Friendship in Edge-Regular Graphs