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 On Distance Integral Graphs Complete Split 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

Things To Know On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs ◮ We assume graphs are simple

Distance Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Definition Distance Matrices Distance Integral Matrices Given a connected graph G on n vertices, the distance Trees Complete Split Graphs matrix D ( G ) is the n x n matrix indexed by the vertex set such that D ( G ) u , v = d G ( u , v ).

Distance Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Definition Distance Matrices Distance Integral Matrices Given a connected graph G on n vertices, the distance Trees Complete Split Graphs matrix D ( G ) is the n x n matrix indexed by the vertex set such that D ( G ) u , v = d G ( u , v ). Example: 0 1 4 3 2

Distance Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Definition Distance Matrices Distance Integral Matrices Given a connected graph G on n vertices, the distance Trees Complete Split Graphs matrix D ( G ) is the n x n matrix indexed by the vertex set such that D ( G ) u , v = d G ( u , v ). Example: 0 1  0 2 2 1 1  4 2 0 1 2 1     D ( G ) = 2 1 0 2 1 3 2     1 2 2 0 1   1 1 1 1 0

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Definition Distance Matrices Distance Integral A graph G is distance integral if its distance spectrum has Matrices Trees only integers. Complete Split Graphs

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Definition Distance Matrices Distance Integral A graph G is distance integral if its distance spectrum has Matrices Trees only integers. Complete Split Graphs Example: 0 5 1 4 6 9 7 8 2 3

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs The Petersen graph is distance integral. (Similar to a potential Exam Question?????)

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs The Petersen graph is distance integral. (Similar to a potential Exam Question?????) ◮ The Petersen graph is r -regular

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs The Petersen graph is distance integral. (Similar to a potential Exam Question?????) ◮ The Petersen graph is r -regular ◮ The Petersen graph has diameter 2

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs 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 ) = 2 J − 2 I − A ( G )

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs 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 ) = 2 J − 2 I − A ( G ) [15 , 0 , 0 , 0 , 0 , − 3 , − 3 , − 3 , − 3 , − 3]

Distance Integral Matrices On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Proposition Trees Complete Split Graphs 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 ) = 2 J − 2 I − A ( G ) [15 , 0 , 0 , 0 , 0 , − 3 , − 3 , − 3 , − 3 , − 3] Or use sage (but don’t for the potential exam question?)

Trees On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Theorem (Merris) Complete Split Graphs Let T be a tree. Then the eigenvalues of − 2( Q T Q ) − 1 ( T ) interlace with the eigenvalues of D ( T ) (where Q = ( q ue ) is the vertex-edge incidence matrix of T such that q ue = 1 if vertex u is the head of edge e, q ue = − 1 if vertex u is the tail of e, and q ue = 0 otherwise).

Trees On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Theorem (Merris) Complete Split Graphs Let T be a tree. Then the eigenvalues of − 2( Q T Q ) − 1 ( T ) interlace with the eigenvalues of D ( T ) (where Q = ( q ue ) is the vertex-edge incidence matrix of T such that q ue = 1 if vertex u is the head of edge e, q ue = − 1 if vertex u is the tail of e, and q ue = 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

Trees On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Corollary (Grone, Merris, Sunder) Complete Split Graphs The number of Laplacian eigenvalues greater than two in a � d � tree T with diameter d is at least . 2

Trees On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Corollary (Grone, Merris, Sunder) Complete Split Graphs The number of Laplacian eigenvalues greater than two in a � d � tree T with diameter d is at least . 2 Corollary (Stevanovi´ c, Indulal) The distance spectrum of the complete bipartite graph K m , n √ m 2 − mn + n 2 consists of simple eigenvalues m + n − 2 ± and an eigenvalue − 2 with multiplicity m + n − 2 . If √ m 2 − mn + n 2 . m , n ≥ 2 , then m + n − 2 ≥

Trees On Distance Integral Graphs Joe Alameda Theorem (Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c) Summary Things to know Every Tree T with at least three vertices has a distance Distance Matrices Distance Integral eigenvalue in the interval ( − 1 , 0) . Matrices Trees Complete Split Graphs

Trees On Distance Integral Graphs Joe Alameda Theorem (Pokorn´ y, H´ ıc, Stevanovi´ c, Milˇ sevi´ c) Summary Things to know Every Tree T with at least three vertices has a distance Distance Matrices Distance Integral eigenvalue in the interval ( − 1 , 0) . Matrices Trees Complete Split Graphs

Proof of Theorem On Distance Integral Graphs Joe Alameda Summary Things to know Distance Matrices Distance Integral Matrices Trees Complete Split Graphs We first note that Q T Q = 2 I + A ( T ∗ ) where A ( T ∗ ) is the adjacency matrix of the line graph of T . We also know that the Laplacian matrix L ( T ) = QQ T has the same non-zero eigenvalues as Q T Q . Now let λ 1 ≥ λ 2 ≥ · · · ≥ λ n be eigenvalues for Q T Q and d 1 ≥ d 2 ≥ · · · ≥ d n be eigenvalues for D ( T ).

On Distance Integral Graphs Joe Alameda Summary Things to know Since the eigenvalues of − 2( Q T Q ) − 1 ( T ) interlace with Distance Matrices Distance Integral D ( T ), Matrices Trees Complete Split Graphs − 2 ≥ d 2 ≥ − 2 λ 1 λ 2 . From the previous theorem we know that the number of Laplacian eigenvalues greater than two in a tree T with � d � diameter d is at least . 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 If T has diameter two, it has a star S n = K 1 , n − 1 . which has √ Distance Matrices n 2 − 3 n + 3 and eigenvalues − 2 with Distance Integral eigenvalues n − 2 ± Matrices Trees multiplicity n − 2. Complete Split Graphs √ √ n 2 − 2 n + 1 < n − 2 − n 2 − 3 n + 3 < − 1 = n − 2 − √ n 2 − 4 n + 4 = 0 n − 2 − So if T has diameter two, it has an eigenvalue in ( − 1 , 0).

On Distance Integral Graphs Joe Alameda Summary Things to know If T has diameter two, it has a star S n = K 1 , n − 1 . which has √ Distance Matrices n 2 − 3 n + 3 and eigenvalues − 2 with Distance Integral eigenvalues n − 2 ± Matrices Trees multiplicity n − 2. Complete Split Graphs √ √ n 2 − 2 n + 1 < n − 2 − n 2 − 3 n + 3 < − 1 = n − 2 − √ n 2 − 4 n + 4 = 0 n − 2 − 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).