On Distance Integral Graphsorion.math.iastate.edu/butler/2017/spring/x95/... · Distance Integral...

Transcript of On Distance Integral Graphsorion.math.iastate.edu/butler/2017/spring/x95/... · Distance Integral...

On DistanceIntegral Graphs

Joe Alameda

Summary

Things to know

Distance Matrices

Distance IntegralMatrices

Trees

Complete Split GraphsOn Distance Integral Graphs

Pokorný, H́ıc, Stevanović, Miľsević

April 14, 2017

Joe Alameda

Summary

Things to know

Distance Matrices


Trees

Complete Split GraphsSummaryThings to knowDistance MatricesDistance Integral MatricesTreesComplete Split Graphs

Joe Alameda

Summary

Things to know

Distance Matrices


Trees

Complete Split Graphs

Things To Know

I We assume graphs are simple

Joe Alameda

Summary

Things to know

Distance Matrices


Trees


Distance Matrices

DefinitionGiven a connected graph G on n vertices, the distancematrix D(G ) is the n x n matrix indexed by the vertex setsuch that D(G )u,v = dG (u, v).

Example:

0 1

23

4

D(G ) =

0 2 2 1 12 0 1 2 12 1 0 2 11 2 2 0 11 1 1 1 0

Joe Alameda

Summary

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Distance Matrices


Trees


Distance Matrices


Example:

0 1

23

4

D(G ) =

0 2 2 1 12 0 1 2 12 1 0 2 11 2 2 0 11 1 1 1 0

Joe Alameda

Summary

Things to know

Distance Matrices


Trees


Distance Matrices


Example:

0 1

23

4

D(G ) =

0 2 2 1 12 0 1 2 12 1 0 2 11 2 2 0 11 1 1 1 0

Joe Alameda

Summary

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Distance Matrices


Trees


Distance Integral Matrices

DefinitionA graph G is distance integral if its distance spectrum hasonly integers.

Example:

0

1

2 3

45

6

7 8

9

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Distance Matrices


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DefinitionA graph G is distance integral if its distance spectrum hasonly integers.

Example:

0

1

2 3

45

6

7 8

9

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Summary

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Distance Matrices


Trees



Proposition

The Petersen graph is distance integral.

(Similar to a potential Exam Question?????)

I The Petersen graph is r -regular

I The Petersen graph has diameter 2

I 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?)

Joe Alameda

Summary

Things to know

Distance Matrices


Trees



Proposition





I D(G ) = 2J − 2I − A(G )

[15, 0, 0, 0, 0,−3,−3,−3,−3,−3]


Joe Alameda

Summary

Things to know

Distance Matrices


Trees



Proposition





I D(G ) = 2J − 2I − A(G )

[15, 0, 0, 0, 0,−3,−3,−3,−3,−3]


Joe Alameda

Summary

Things to know

Distance Matrices


Trees



Proposition





I D(G ) = 2J − 2I − A(G )

[15, 0, 0, 0, 0,−3,−3,−3,−3,−3]


Joe Alameda

Summary

Things to know

Distance Matrices


Trees



Proposition





I D(G ) = 2J − 2I − A(G )

[15, 0, 0, 0, 0,−3,−3,−3,−3,−3]


Joe Alameda

Summary

Things to know

Distance Matrices


Trees



Proposition





I D(G ) = 2J − 2I − A(G )

[15, 0, 0, 0, 0,−3,−3,−3,−3,−3]


Joe Alameda

Summary

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Distance Matrices


Trees


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) isthe vertex-edge incidence matrix of T such that que = 1 ifvertex u is the head of edge e, que = −1 if vertex u is thetail of e, and que = 0 otherwise).

Proof.Obvious according to the paper this theorem is in.

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Distance Matrices


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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) isthe vertex-edge incidence matrix of T such that que = 1 ifvertex u is the head of edge e, que = −1 if vertex u is thetail of e, and que = 0 otherwise).

Proof.Obvious according to the paper this theorem is in.

Joe Alameda

Summary

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Distance Matrices


Trees


Joe Alameda

Summary

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Distance Matrices


Trees


Trees

Corollary (Grone, Merris, Sunder)

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

⌊d2

⌋.

Corollary (Stevanović, Indulal)

The distance spectrum of the complete bipartite graph Km,nconsists of simple eigenvalues m + n − 2±

√m2 −mn + n2

and an eigenvalue −2 with multiplicity m + n − 2. Ifm, n ≥ 2, then m + n − 2 ≥

√m2 −mn + n2.

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Summary

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Distance Matrices


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Trees

Corollary (Grone, Merris, Sunder)

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

⌊d2

⌋.

Corollary (Stevanović, Indulal)

The distance spectrum of the complete bipartite graph Km,nconsists of simple eigenvalues m + n − 2±

√m2 −mn + n2

and an eigenvalue −2 with multiplicity m + n − 2. Ifm, n ≥ 2, then m + n − 2 ≥

√m2 −mn + n2.

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Summary

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Distance Matrices


Trees


Trees

Theorem (Pokorný, H́ıc, Stevanović, Miľsević)

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

Joe Alameda

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Distance Matrices


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Trees


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

Joe Alameda

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Distance Matrices


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Proof of Theorem

We first note that QTQ = 2I + A(T ∗) where A(T ∗) is theadjacency matrix of the line graph of T .We also know that the Laplacian matrix L(T ) = QQT hasthe same non-zero eigenvalues as QTQ.Now let λ1 ≥ λ2 ≥ · · · ≥ λn be eigenvalues for QTQ andd1 ≥ d2 ≥ · · · ≥ dn be eigenvalues for D(T ).

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Distance Matrices


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Since the eigenvalues of −2(QTQ)−1(T ) interlace withD(T ),

− 2λ1≥ d2 ≥ −

2

λ2.

From the previous theorem we know that the number ofLaplacian eigenvalues greater than two in a tree T withdiameter d is at least

⌊d2

⌋.

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

Joe Alameda

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Distance Matrices


Trees


If T has diameter two, it has a star Sn = K1,n−1. which haseigenvalues 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 hasan eigenvalue in (−1, 0).

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Distance Matrices


Trees


If T has diameter two, it has a star Sn = K1,n−1. which haseigenvalues 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 hasan eigenvalue in (−1, 0).

Joe Alameda

Summary

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Distance Matrices


Trees


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

Joe Alameda

Summary

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Distance Matrices


Trees


Complete Spit Graphs

DefinitionThe join G1OG2 of graphs G1 and G2 is a graph obtainedfrom the union of G1 and G2 by adding an edge joining everyvertex of G1 to every vertex of G2.

DefinitionFor a, b, n ∈ N we define the complete split graphCSab = K̄aOKb

the multiple complete split-like graphMCSab,n = K̄aOnKb the multiple extended complete split-like

graph ECSab,n = K̄aO(Kb + K2) and the multiple extended

complete split-like graph MECSab,n = K̄aOn(Kb + K2)

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DefinitionFor a, b, n ∈ N we define the complete split graphCSab = K̄aOKb the multiple complete split-like graphMCSab,n = K̄aOnKb

the multiple extended complete split-like



Joe Alameda

Summary

Things to know

Distance Matrices


Trees




DefinitionFor a, b, n ∈ N we define the complete split graphCSab = K̄aOKb the multiple complete split-like graphMCSab,n = K̄aOnKb the multiple extended complete split-like

graph ECSab,n = K̄aO(Kb + K2)

and the multiple extended


Joe Alameda

Summary

Things to know

Distance Matrices


Trees




DefinitionFor a, b, n ∈ N we define the complete split graphCSab = K̄aOKb the multiple complete split-like graphMCSab,n = K̄aOnKb the multiple extended complete split-like



Joe Alameda

Summary

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Distance Matrices


Trees



Theorem (Stevanović, Indulal)

For i = 1, 2, let Gi be an ri -regular graph with ni verticesand the eigenvalues λi ,1 = ri ≥ · · · ≥ λi ,n of the adjacencymatrix of Gi . The distance spectrum of G1OG2 consists ofthe eigenvalues −λi ,j − 2 for i = 1, 2 and j = 2, 3, · · · , niand two more simple eigenvalues

n1 + n2 − 2−r1 + r2

2±√

(n1 − n2 −r1 − r2

2)2 + n1n2.

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Distance Matrices


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Joe Alameda

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Distance Matrices


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We will prove the first part.

Proof.K̄a is 0-regular and Kb is b − 1-regular. The spectrum of theadjacency matrix of K̄a is 0

a and the spectrum of theadjacency matrix of Kb is b − 1 and −1b−1. By the lasttheorem K̄aOKb has eigenvalues −2a−1, −1b−1 and twosimple eigenvalues

2a + b − 32

±√

4a(a− 1) + (b + 1)22

.

Since 2a + b − 3 and 4a(a− 1) + (b + 1)2 are integers withthe same parity, the above is an integer if and only if4a(a− 1) + (b + 1)2 is a perfect square.

SummaryThings to knowDistance MatricesDistance Integral MatricesTreesComplete Split Graphs

On Distance Integral Graphsorion.math.iastate.edu/butler/2017/spring/x95/... · Distance Integral...

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Transcript of On Distance Integral Graphsorion.math.iastate.edu/butler/2017/spring/x95/... · Distance Integral...