Maximum Laplacian Energy among Threshold Graphs Christoph Helmberg - - PowerPoint PPT Presentation

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Maximum Laplacian Energy among Threshold Graphs

Christoph Helmberg (TU Chemnitz) joint work with Vilmar Trevisan (UFRGS, Porto Alegre, Brasil)

• Laplacian Energy (LE)
• Threshold Graphs
• Ferrers Diagrams
• Maximal LE(TG) for fixed n, m, f
• Maximal LE(TG) for fixed n and m
• Maximal LE(TG) for fixed n
• The connected case
• Extensions and Open Problems

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

The Laplacian Energy of a graph

• finite simple undirected Graph G = (V , E),

n nodes V = {1, . . . , n}, m edges E ⊆ {{i, j} : i, j ∈ V , i = j} [ij ∈ E]

3 4 2 1

• Laplacian [L(G)]ij =

   deg(i) i = j −1 ij ∈ E

• therwise

L =     2 −1 −1 −1 2 −1 −1 −1 3 −1 0 −1 1    

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

The Laplacian Energy of a graph

• finite simple undirected Graph G = (V , E),

n nodes V = {1, . . . , n}, m edges E ⊆ {{i, j} : i, j ∈ V , i = j} [ij ∈ E]

3 4 2 1

• Laplacian [L(G)]ij =

   deg(i) i = j −1 ij ∈ E

• therwise

L =     2 −1 −1 −1 2 −1 −1 −1 3 −1 0 −1 1    

• L(G) =
• ij∈E

Eij Eij =

• 1

−1 −1 1 i j

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

The Laplacian Energy of a graph

• finite simple undirected Graph G = (V , E),

n nodes V = {1, . . . , n}, m edges E ⊆ {{i, j} : i, j ∈ V , i = j} [ij ∈ E]

3 4 2 1

• Laplacian [L(G)]ij =

   deg(i) i = j −1 ij ∈ E

• therwise

L =     2 −1 −1 −1 2 −1 −1 −1 3 −1 0 −1 1    

• L(G) =
• ij∈E

Eij Eij =

• 1

−1 −1 1 i j = (ei − ej)(ei − ej)T

• L is pos. semidef., eigenvalues λ1 ≥ · · · ≥ λn−1 ≥ λn = 0.
• average degree ¯

δ = 2m/n,

• i∈V λi = 2m = n¯

δ

• Laplacian Energy LE(G) =

i∈V |λi − ¯

δ| Given n, which (connected) graphs maximize the Laplacian Energy?

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Pineapple Conjecture

On n nodes a connected graph maximizing the Laplacian energy is the pineapple graph (a clique on {1, . . . , ⌊ 2n

3 ⌋ + 1} plus edges {{1, i}: i = ⌊ 2n 3 ⌋ + 2, . . . , n}).

• V. Trevisan

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Pineapple Conjecture

On n nodes a connected graph maximizing the Laplacian energy is the pineapple graph (a clique on {1, . . . , ⌊ 2n

3 ⌋ + 1} plus edges {{1, i}: i = ⌊ 2n 3 ⌋ + 2, . . . , n}).

• V. Trevisan

We cannot prove this conjecture, but the pineapple is a threshold graph and we can prove it to be a maximizer over all threshold graphs. For not necessarily connected graphs we are able to exhibit maximimizers

• ver all threshold graphs, split graphs, and cographs.

We only discuss the threshold case here.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Threshold Graphs [MahadevPeled1995]

first introduced by Chv´ atal and Hammer for independent sets G is a threshold graph if it can be constructed from the one-vertex graph by repeatedly adding an isolated vertex or a dominating vertex (=connected to all previous ones). expressed via a 0-1 sequence 0 1 1 1 1 0 0 1

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Threshold Graphs [MahadevPeled1995]

first introduced by Chv´ atal and Hammer for independent sets G is a threshold graph if it can be constructed from the one-vertex graph by repeatedly adding an isolated vertex or a dominating vertex (=connected to all previous ones). expressed via a 0-1 sequence 0 1 1 1 1 0 0 1 degree sequence: 7 5 5 5 5 5 1 1

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Threshold Graphs [MahadevPeled1995]

first introduced by Chv´ atal and Hammer for independent sets G is a threshold graph if it can be constructed from the one-vertex graph by repeatedly adding an isolated vertex or a dominating vertex (=connected to all previous ones). expressed via a 0-1 sequence 0 1 1 1 1 0 0 1 degree sequence: 7 5 5 5 5 5 1 1 Denote by d1 ≥ d2 ≥ · · · ≥ dn the nonincreasing degree sequence of G, then f = max{j : dj ≥ j} is called its trace, and d∗

i = max{j : dj ≥ i}, i = 1, . . . , n

its dual degree sequence. A graph is threshold if and only if d∗

i = di + 1 for i = 1, . . . , f .

→ Th(d∗).

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Ferrers (or Young) diagram

illustrates a degree sequence by rows of boxes, e.g. for d=(7,6,6,5,4,4,3,1) [n = 8, m = 17, ¯ δ = 36/8] d1 d2 d3 d4 d5 d6 d7 d8 d∗

1 d∗ 2 d∗ 3 d∗ 4 d∗ 5 d∗ 6 d∗ 7 d∗ 8

trace = number of black boxes [f = 4] Diagrams

• f

threshold graphs are “symmetric”. Note: d∗

i ≤ f

for i > f , d∗

i ≥ f + 1 for i ≤ f .

!! For threshold graphs λi = d∗

i , i = 1, . . . , n !!

[Merris94]

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Laplacian Energy of Threshold Graphs

LE(Th(d∗)) =

• i∈V

|d∗

i − ¯

δ| =

• d∗

i >¯

δ

(d∗

i − ¯

δ) +

• d∗

i ≤¯

δ

(¯ δ − d∗

i )

¯ δ 1 2 3 4 5 6 7 8

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Laplacian Energy of Threshold Graphs

LE(Th(d∗)) =

• i∈V

|d∗

i − ¯

δ| =

• d∗

i >¯

δ

(d∗

i − ¯

δ) +

• d∗

i ≤¯

δ

(¯ δ − d∗

i )

Note,

i∈V (d∗ i − ¯

δ) = 0, ¯ δ 1 2 3 4 5 6 7 8

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Laplacian Energy of Threshold Graphs

LE(Th(d∗)) =

• i∈V

|d∗

i − ¯

δ| =

• d∗

i >¯

δ

(d∗

i − ¯

δ) +

• d∗

i ≤¯

δ

(¯ δ − d∗

i )

Note,

i∈V (d∗ i − ¯

δ) = 0, thus

d∗

i >¯

δ(d∗ i − ¯

δ) =

d∗

i ≤¯

δ(¯

δ − d∗

i )

¯ δ 1 2 3 4 5 6 7 8

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Laplacian Energy of Threshold Graphs

LE(Th(d∗)) =

• i∈V

|d∗

i − ¯

δ| =

• d∗

i >¯

δ

(d∗

i − ¯

δ) +

• d∗

i ≤¯

δ

(¯ δ − d∗

i )

Note,

i∈V (d∗ i − ¯

δ) = 0, thus

d∗

i >¯

δ(d∗ i − ¯

δ) =

d∗

i ≤¯

δ(¯

δ − d∗

i ) and

LE(Th(d∗)) = 2

• d∗

i ≥¯

δ

(d∗

i − ¯

δ) = 2

• d∗

i ≤¯

δ

(¯ δ − d∗

i )

¯ δ 1 2 3 4 5 6 7 8

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Increase LE for fixed n, m, f by shifting boxes

For fixed f and f ≤ ¯ δ ≤ f + 1, all symmetric box arrangements are fine:

1 2 3 4 5 6 7 8

arbitrary =

1 2 3 4 5 6 7 8

lexmin degree =

1 2 3 4 5 6 7 8

lexmax degree

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Increase LE for fixed n, m, f by shifting boxes

For fixed f and f ≤ ¯ δ ≤ f + 1, all symmetric box arrangements are fine:

1 2 3 4 5 6 7 8

arbitrary =

1 2 3 4 5 6 7 8

lexmin degree =

1 2 3 4 5 6 7 8

lexmax degree If ¯ δ < f , lexmin will be an optimal arrangement:

1 2 3 4 5 6 7 8

arbitrary ≤

1 2 3 4 5 6 7 8

lexmin degree

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Increase LE for fixed n, m, f by shifting boxes

For fixed f and f ≤ ¯ δ ≤ f + 1, all symmetric box arrangements are fine:

1 2 3 4 5 6 7 8

arbitrary =

1 2 3 4 5 6 7 8

lexmin degree =

1 2 3 4 5 6 7 8

lexmax degree If ¯ δ < f , lexmin will be an optimal arrangement:

1 2 3 4 5 6 7 8

arbitrary ≤

1 2 3 4 5 6 7 8

lexmin degree If ¯ δ > f + 1, lexmax will be an optimal arrangement (examples are big).

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Lemma

Among all threshold graphs on n nodes and m edges with degree sequence

• f trace f the maximum LE is attained for
• the one having lexmin degree sequence if ¯

δ ≤ f + 1,

• the one having lexmax degree sequence if ¯

δ ≥ f + 1.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Maximum LE for Threshold Graphs with fixed n, m

For fixed n and m, feasible trace value f satisfy f (f + 1) ≤ 2m and f (f + 1) + 2(n − 1 − f )f ≥ 2m resulting in upper and lower bounds on f , f :=

• n − 1

2 −

• n2 − n + 1

4 − 2m

• ≤ f ≤
• −1

2 +

• 2m + 1

4

• =: ¯

f . Example for n = 7, m = 11

1 2 3 4 5 6 7

lexmin for f = 3

1 2 3 4 5 6 7

lexmax for f = 2

1 2 3 4 5 6 7

lexmin for ¯ f = 4

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n and m a threshold graph of maximum LE can be found among the graphs lexmax for f and lexmin for ¯ f .

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n and m a threshold graph of maximum LE can be found among the graphs lexmax for f and lexmin for ¯ f .

Proof.

Let T be a thresholdgraph of maxium LE for n and m, let it have trace f . Case ¯ δ ≤ f + 1: Case ¯ δ > f + 1:

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n and m a threshold graph of maximum LE can be found among the graphs lexmax for f and lexmin for ¯ f .

Proof.

Let T be a thresholdgraph of maxium LE for n and m, let it have trace f . Case ¯ δ ≤ f + 1: Lemma ⇒ we may assume T is lexmin for this trace. If f = ¯ f we are done. Case ¯ δ > f + 1:

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n and m a threshold graph of maximum LE can be found among the graphs lexmax for f and lexmin for ¯ f .

Proof.

Let T be a thresholdgraph of maxium LE for n and m, let it have trace f . Case ¯ δ ≤ f + 1: Lemma ⇒ we may assume T is lexmin for this trace. If f = ¯ f we are done. Otherwise we may increase f and LE by shifting a box like in

1 2 3 4 5 6 7

lexmin for f = 3

1 2 3 4 5 6 7

lexmin for ¯ f = 4 Case ¯ δ > f + 1:

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n and m a threshold graph of maximum LE can be found among the graphs lexmax for f and lexmin for ¯ f .

Proof.

Let T be a thresholdgraph of maxium LE for n and m, let it have trace f . Case ¯ δ ≤ f + 1: Lemma ⇒ we may assume T is lexmin for this trace. If f = ¯ f we are done. Otherwise we may increase f and LE by shifting a box like in

1 2 3 4 5 6 7

lexmin for f = 3

1 2 3 4 5 6 7

lexmin for ¯ f = 4 Case ¯ δ > f + 1: same argument leads to lexmax for f .

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

LE Formulas of lexmax for f and lexmin for ¯ f

Given n and m, the Laplacian Energy of lexmax for f (conj. degs. d∗) reads TE(n, m) := LE(Th(d∗)) = 2¯ δ(n − m) 2m ≤ n, 2[(f − 1)(n − ¯ δ) + max{0, d∗

f − ¯

δ}] 2m ≥ n, with d∗

f = f + 1 + m − f (f + 1)/2 − (f − 1)(n − 1 − f ).

and the Laplacian Energy of lexmin for ¯ f (conj. degs. ¯ d∗) reads TE(n, m) := LE(Th(¯ d∗)) = 2[(n − 1 − ¯ f )¯ δ + max{0, ¯ δ − ¯ d∗

¯ f +1}]

with ¯ d∗

¯ f +1 = m − ¯

f (¯ f + 1)/2.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Maximum LE for Threshold Graphs with fixed n

Path:

• Study TE(n, m) (lexmax for f ) for m = 1, . . . ,

n

2

• → best m.
• Study TE(n, m) (lexmin for ¯

f ) for m = 1, . . . , n

2

• → best ¯

m.

• Compare the two.

We illustrate the idea of the proof for TE(n, m) (lexmin).

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box kills 1 unit, but adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box kills 1 unit, but adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box kills 1 unit, but adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box kills 1 unit, but adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box kills part of a unit, but adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box adds 2

n to ¯

δ

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box adds 2

n to ¯

δ In particular, if adding the first box of a column increases LE, all boxes in this column increase LE.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box adds 2

n to ¯

δ In particular, if adding the first box of a column increases LE, all boxes in this column increase LE. → we only need to compare LE for “full rectangles” (cliques).

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box adds 2

n to ¯

δ In particular, if adding the first box of a column increases LE, all boxes in this column increase LE. → we only need to compare LE for “full rectangles” (cliques).

• For m = k(k + 1)/2, k = 1, . . . , n − 1 the formula simplifies to

n 2TE(n, k(k + 1)/2) = nk(k + 1 − ¯

δ) = (n − k)k(k + 1). and is maximized for k = ⌊ 1

3(2n + 1)⌋.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

• TE(n, .) is “convex” along columns:

1 2 3 4 5 6 7

the new box adds 2

n to ¯

δ In particular, if adding the first box of a column increases LE, all boxes in this column increase LE. → we only need to compare LE for “full rectangles” (cliques).

• For m = k(k + 1)/2, k = 1, . . . , n − 1 the formula simplifies to

n 2TE(n, k(k + 1)/2) = nk(k + 1 − ¯

δ) = (n − k)k(k + 1). and is maximized for k = ⌊ 1

3(2n + 1)⌋.

Similar arguments show that TE(n, ·) is maximized for ¯ m = n − 1 + ¯ k(¯ k − 1)/2 with ¯ k = ⌊ 2

3n⌋.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n ≥ 2 a threshold graph on n nodes maximizing the Laplacian energy is the lexmin graph having conjugate degree sequence d∗

i = k + 1,

i ∈ [k], and d∗

k+i = 0, i ∈ [n − k], with trace k = ⌊ 1 3(2n + 1)⌋.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n ≥ 2 a threshold graph on n nodes maximizing the Laplacian energy is the lexmin graph having conjugate degree sequence d∗

i = k + 1,

i ∈ [k], and d∗

k+i = 0, i ∈ [n − k], with trace k = ⌊ 1 3(2n + 1)⌋.

Proof.

Use ¯ k = ⌊ 2

3n⌋ and k = ⌊ 1 3(2n + 1)⌋, these give rise to ¯

m, m. Showing TE(n, m) ≤ TE(n, ¯ m) simplifies to proving k3 + (1 − 2n)k2 + n2k ≤ (n − ¯ k)¯ k(¯ k + 1).

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Theorem

For given n ≥ 2 a threshold graph on n nodes maximizing the Laplacian energy is the lexmin graph having conjugate degree sequence d∗

i = k + 1,

i ∈ [k], and d∗

k+i = 0, i ∈ [n − k], with trace k = ⌊ 1 3(2n + 1)⌋.

Proof.

Use ¯ k = ⌊ 2

3n⌋ and k = ⌊ 1 3(2n + 1)⌋, these give rise to ¯

m, m. Showing TE(n, m) ≤ TE(n, ¯ m) simplifies to proving k3 + (1 − 2n)k2 + n2k ≤ (n − ¯ k)¯ k(¯ k + 1). Case 1: n = 3h+1 with h ∈ N → k = h and ¯ k = 2h, left = 4h3 + h2 while right = 4h3 + 2h2. Case 2: n = 3h + 1 with h ∈ N → k = h + 1 and ¯ k = 2h + 1, left = 4h3 + 5h2 + 2h + 1 while right = 4h3 + 6h2 + 2h. Case 3: n = 3h − 1 with h ∈ N → k = h and ¯ k = 2h − 1, left = 4h3 − 3h2 + h while right = 4h3 − 2h2

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

The connected case

For connectedness the first row of Ferrers diagram must be full (node 1 is connected to all others). → lexmax is identical, the arguments for lexmin have to be adapted a bit. The same steps lead directly to

Theorem

For given n ≥ 2 a connected threshold graph on n nodes maximizing the Laplacian energy has conjugate degree sequence d∗

1 = n, d∗ i = k + 1 for

i ∈ {2, . . . , k}, d∗

i = 1 for i ∈ {k + 1, . . . , n − 1} and d∗ n = 0 with

k = ⌊ 2

3n⌋.

This is exactly the pineapple.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Further results, connections, and open problems

• We can prove that for n ≥ 2 the ⌊ 2

3n + 4 3⌋-clique has maximum LE

among all split graphs and cographs and, more generally, among all “spectrally threshold dominated graphs”. G = (V , E) is s.th.d. if for k = 1, . . . , n − 1 there is a threshold graph Tk on |V | nodes and |E| edges so that k

i=1 λi(G) ≤ k i=1 λi(Tk)

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Further results, connections, and open problems

• We can prove that for n ≥ 2 the ⌊ 2

3n + 4 3⌋-clique has maximum LE

among all split graphs and cographs and, more generally, among all “spectrally threshold dominated graphs”. G = (V , E) is s.th.d. if for k = 1, . . . , n − 1 there is a threshold graph Tk on |V | nodes and |E| edges so that k

i=1 λi(G) ≤ k i=1 λi(Tk)

• The conjecture that all graphs are spectrally threshold dominated is

equivalent to the following conjecture of A. E. Brouwer: for all graphs k

i=1 λi(G) ≤ |E| + k(k + 1)/2 for k = 1, . . . , n.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Further results, connections, and open problems

• We can prove that for n ≥ 2 the ⌊ 2

3n + 4 3⌋-clique has maximum LE

among all split graphs and cographs and, more generally, among all “spectrally threshold dominated graphs”. G = (V , E) is s.th.d. if for k = 1, . . . , n − 1 there is a threshold graph Tk on |V | nodes and |E| edges so that k

i=1 λi(G) ≤ k i=1 λi(Tk)

• The conjecture that all graphs are spectrally threshold dominated is

equivalent to the following conjecture of A. E. Brouwer: for all graphs k

i=1 λi(G) ≤ |E| + k(k + 1)/2 for k = 1, . . . , n.

• Spectral threshold dominance is maybe stronger than needed.

introduction TG LE fixed n, m, f fixed n, m fixed n connected conclusion

Further results, connections, and open problems

• We can prove that for n ≥ 2 the ⌊ 2

3n + 4 3⌋-clique has maximum LE

among all split graphs and cographs and, more generally, among all “spectrally threshold dominated graphs”. G = (V , E) is s.th.d. if for k = 1, . . . , n − 1 there is a threshold graph Tk on |V | nodes and |E| edges so that k

i=1 λi(G) ≤ k i=1 λi(Tk)

• The conjecture that all graphs are spectrally threshold dominated is

equivalent to the following conjecture of A. E. Brouwer: for all graphs k

i=1 λi(G) ≤ |E| + k(k + 1)/2 for k = 1, . . . , n.

• Spectral threshold dominance is maybe stronger than needed.
• What can be done for the connected case?