On Distance Integral Graphs Complete Split Graphs Pokorn y, H c, - - PowerPoint PPT Presentation

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

On Distance Integral Graphs

Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c

April 14, 2017

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Summary Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Things To Know

◮ We assume graphs are simple

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Matrices

Definition

Given a connected graph G on n vertices, the distance matrix D(G) is the n x n matrix indexed by the vertex set such that D(G)u,v = dG(u, v).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Matrices

Definition

Given a connected graph G on n vertices, the distance matrix D(G) is the n x n matrix indexed by the vertex set such that D(G)u,v = dG(u, v). Example: 1 2 3 4

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Matrices

Definition

Given a connected graph G on n vertices, the distance matrix D(G) is the n x n matrix indexed by the vertex set such that D(G)u,v = dG(u, v). Example: 1 2 3 4 D(G) =       2 2 1 1 2 1 2 1 2 1 2 1 1 2 2 1 1 1 1 1      

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Definition

A graph G is distance integral if its distance spectrum has

• nly integers.

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Definition

A graph G is distance integral if its distance spectrum has

• nly integers.

Example: 1 2 3 4 5 6 7 8 9

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

◮ The Petersen graph is r-regular

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

◮ The Petersen graph is r-regular ◮ The Petersen graph has diameter 2

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

◮ The Petersen graph is r-regular ◮ The Petersen graph has diameter 2 ◮ D(G) = 2J − 2I − A(G)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

◮ The Petersen graph is r-regular ◮ The Petersen graph has diameter 2 ◮ D(G) = 2J − 2I − A(G)

[15, 0, 0, 0, 0, −3, −3, −3, −3, −3]

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Distance Integral Matrices

Proposition

The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

◮ The Petersen graph is r-regular ◮ The Petersen graph has diameter 2 ◮ D(G) = 2J − 2I − A(G)

[15, 0, 0, 0, 0, −3, −3, −3, −3, −3] Or use sage (but don’t for the potential exam question?)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Trees

Theorem (Merris)

Let T be a tree. Then the eigenvalues of −2(QTQ)−1(T) interlace with the eigenvalues of D(T) (where Q = (que) is the vertex-edge incidence matrix of T such that que = 1 if vertex u is the head of edge e, que = −1 if vertex u is the tail of e, and que = 0 otherwise).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Trees

Theorem (Merris)

Let T be a tree. Then the eigenvalues of −2(QTQ)−1(T) interlace with the eigenvalues of D(T) (where Q = (que) is the vertex-edge incidence matrix of T such that que = 1 if vertex u is the head of edge e, que = −1 if vertex u is the tail of e, and que = 0 otherwise).

Proof.

Obvious according to the paper this theorem is in.

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Trees

Corollary (Grone, Merris, Sunder)

The number of Laplacian eigenvalues greater than two in a tree T with diameter d is at least d

2

• .

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Trees

Corollary (Grone, Merris, Sunder)

The number of Laplacian eigenvalues greater than two in a tree T with diameter d is at least d

2

• .

Corollary (Stevanovi´ c, Indulal)

The distance spectrum of the complete bipartite graph Km,n consists of simple eigenvalues m + n − 2 ± √ m2 − mn + n2 and an eigenvalue −2 with multiplicity m + n − 2. If m, n ≥ 2, then m + n − 2 ≥ √ m2 − mn + n2.

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Trees

Theorem (Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c)

Every Tree T with at least three vertices has a distance eigenvalue in the interval (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Trees

Theorem (Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c)

Every Tree T with at least three vertices has a distance eigenvalue in the interval (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Proof of Theorem

We first note that QTQ = 2I + A(T ∗) where A(T ∗) is the adjacency matrix of the line graph of T. We also know that the Laplacian matrix L(T) = QQT has the same non-zero eigenvalues as QTQ. Now let λ1 ≥ λ2 ≥ · · · ≥ λn be eigenvalues for QTQ and d1 ≥ d2 ≥ · · · ≥ dn be eigenvalues for D(T).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Since the eigenvalues of −2(QTQ)−1(T) interlace with D(T), − 2 λ1 ≥ d2 ≥ − 2 λ2. From the previous theorem we know that the number of Laplacian eigenvalues greater than two in a tree T with diameter d is at least d

2

• .

Therefore by the inequality above, for any tree with diameter at least four there exists an eigenvalue in (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

If T has diameter two, it has a star Sn = K1,n−1. which has eigenvalues n − 2 ± √ n2 − 3n + 3 and eigenvalues −2 with multiplicity n − 2. −1 = n − 2 − √ n2 − 2n + 1 < n − 2 − √ n2 − 3n + 3 < n − 2 − √ n2 − 4n + 4 = 0 So if T has diameter two, it has an eigenvalue in (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

If T has diameter two, it has a star Sn = K1,n−1. which has eigenvalues n − 2 ± √ n2 − 3n + 3 and eigenvalues −2 with multiplicity n − 2. −1 = n − 2 − √ n2 − 2n + 1 < n − 2 − √ n2 − 3n + 3 < n − 2 − √ n2 − 4n + 4 = 0 So if T has diameter two, it has an eigenvalue in (−1, 0). If T is a path on four vertices it has diameter three and has an eigenvalue in (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

If T has diameter three and is not a path on four vertices, it has a subgraph (draw picture). Since the Laplacian eigenvalues interlace and if e is an edge

• f a graph G, then

λi(G) ≥ λi(G − e) ≥ λi+1(G) for i = 1, · · · , n − 1. Since the graph F can be obtained by deleting edges from T, λ2 ≥ λ2(F). Using our first inequality we get that d2 ∈ (−1, 0).

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Spit Graphs

Definition

The join G1▽G2 of graphs G1 and G2 is a graph obtained from the union of G1 and G2 by adding an edge joining every vertex of G1 to every vertex of G2.

Definition

For a, b, n ∈ N we define the complete split graph CSa

b = ¯

Ka▽Kb

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Spit Graphs

Definition

The join G1▽G2 of graphs G1 and G2 is a graph obtained from the union of G1 and G2 by adding an edge joining every vertex of G1 to every vertex of G2.

Definition

For a, b, n ∈ N we define the complete split graph CSa

b = ¯

Ka▽Kb the multiple complete split-like graph MCSa

b,n = ¯

Ka▽nKb

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Spit Graphs

Definition

The join G1▽G2 of graphs G1 and G2 is a graph obtained from the union of G1 and G2 by adding an edge joining every vertex of G1 to every vertex of G2.

Definition

For a, b, n ∈ N we define the complete split graph CSa

b = ¯

Ka▽Kb the multiple complete split-like graph MCSa

b,n = ¯

Ka▽nKb the multiple extended complete split-like graph ECSa

b,n = ¯

Ka▽(Kb + K2)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Spit Graphs

Definition

The join G1▽G2 of graphs G1 and G2 is a graph obtained from the union of G1 and G2 by adding an edge joining every vertex of G1 to every vertex of G2.

Definition

For a, b, n ∈ N we define the complete split graph CSa

b = ¯

Ka▽Kb the multiple complete split-like graph MCSa

b,n = ¯

Ka▽nKb the multiple extended complete split-like graph ECSa

b,n = ¯

Ka▽(Kb + K2) and the multiple extended complete split-like graph MECSa

b,n = ¯

Ka▽n(Kb + K2)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Split Graphs

Theorem (Stevanovi´ c, Indulal)

For i = 1, 2, let Gi be an ri-regular graph with ni vertices and the eigenvalues λi,1 = ri ≥ · · · ≥ λi,n of the adjacency matrix of Gi. The distance spectrum of G1▽G2 consists of the eigenvalues −λi,j − 2 for i = 1, 2 and j = 2, 3, · · · , ni and two more simple eigenvalues n1 + n2 − 2 − r1 + r2 2 ±

• (n1 − n2 − r1 − r2

2 )2 + n1n2.

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Split Graphs

Theorem (Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c)

On Distance Integral Graphs Joe Alameda Summary

Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs

Complete Split Graphs

We will prove the first part.

Proof.

¯ Ka is 0-regular and Kb is b − 1-regular. The spectrum of the adjacency matrix of ¯ Ka is 0a and the spectrum of the adjacency matrix of Kb is b − 1 and −1b−1. By the last theorem ¯ Ka▽Kb has eigenvalues −2a−1, −1b−1 and two simple eigenvalues 2a + b − 3 2 ±

• 4a(a − 1) + (b + 1)2

2 . Since 2a + b − 3 and 4a(a − 1) + (b + 1)2 are integers with the same parity, the above is an integer if and only if 4a(a − 1) + (b + 1)2 is a perfect square.