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 1 • finite simple undirected Graph G = ( V , E ), n nodes V = { 1 , . . . , n } , 3 4 m edges E ⊆ {{ i , j } : i , j ∈ V , i � = j } [ ij ∈ E ] 2   2 − 1 − 1 0  deg ( i ) i = j − 1 2 − 1 0    • Laplacian [ L ( G )] ij = − 1 ij ∈ E L =   − 1 − 1 3 − 1   0 otherwise  0 0 − 1 1

introduction TG LE fixed n , m , f fixed n , m fixed n connected conclusion The Laplacian Energy of a graph 1 • finite simple undirected Graph G = ( V , E ), n nodes V = { 1 , . . . , n } , 3 4 m edges E ⊆ {{ i , j } : i , j ∈ V , i � = j } [ ij ∈ E ] 2   2 − 1 − 1 0  deg ( i ) i = j − 1 2 − 1 0    • Laplacian [ L ( G )] ij = − 1 ij ∈ E L =   − 1 − 1 3 − 1   0 otherwise  0 0 − 1 1 � i � 1 − 1 � • L ( G ) = E ij = E ij − 1 1 j ij ∈ E

introduction TG LE fixed n , m , f fixed n , m fixed n connected conclusion The Laplacian Energy of a graph 1 • finite simple undirected Graph G = ( V , E ), n nodes V = { 1 , . . . , n } , 3 4 m edges E ⊆ {{ i , j } : i , j ∈ V , i � = j } [ ij ∈ E ] 2   2 − 1 − 1 0  deg ( i ) i = j − 1 2 − 1 0    • Laplacian [ L ( G )] ij = − 1 ij ∈ E L =   − 1 − 1 3 − 1   0 otherwise  0 0 − 1 1 � i � 1 − 1 � = ( e i − e j )( e i − e j ) T • L ( G ) = E ij = E ij − 1 1 j ij ∈ E • L is pos. semidef., eigenvalues λ 1 ≥ · · · ≥ λ n − 1 ≥ λ n = 0. • average degree ¯ i ∈ V λ i = 2 m = n ¯ δ = 2 m / n , � δ i ∈ V | λ i − ¯ • Laplacian Energy LE ( G ) = � 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 , . . . , ⌊ 2 n 3 ⌋ + 1 } plus edges {{ 1 , i } : i = ⌊ 2 n 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 , . . . , ⌊ 2 n 3 ⌋ + 1 } plus edges {{ 1 , i } : i = ⌊ 2 n 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 over 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 d 1 ≥ d 2 ≥ · · · ≥ d n the nonincreasing degree sequence of G , then f = max { j : d j ≥ j } is called its trace, and d ∗ i = max { j : d j ≥ i } , i = 1 , . . . , n its dual degree sequence. A graph is threshold if and only if d ∗ i = d i + 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, [ n = 8 , m = 17 , ¯ e.g. for d=(7,6,6,5,4,4,3,1) δ = 36 / 8] d ∗ 1 d ∗ 2 d ∗ 3 d ∗ 4 d ∗ 5 d ∗ 6 d ∗ 7 d ∗ 8 trace = number of black boxes [ f = 4] d 1 d 2 Diagrams of threshold graphs are d 3 “symmetric”. d 4 Note: d ∗ i ≤ f for i > f , d 5 d ∗ i ≥ f + 1 for i ≤ f . d 6 d 7 d 8 !! For threshold graphs λ i = 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 � i − ¯ � i − ¯ � (¯ LE ( Th ( d ∗ )) = | d ∗ ( d ∗ δ − d ∗ δ | = δ ) + i ) i ∈ V i > ¯ i ≤ ¯ d ∗ δ d ∗ δ 0 1 2 3 4 ¯ δ 5 6 7 8

introduction TG LE fixed n , m , f fixed n , m fixed n connected conclusion Laplacian Energy of Threshold Graphs � i − ¯ � i − ¯ � (¯ LE ( Th ( d ∗ )) = | d ∗ ( d ∗ δ − d ∗ δ | = δ ) + i ) i ∈ V i > ¯ i ≤ ¯ d ∗ δ d ∗ δ i − ¯ Note, � i ∈ V ( d ∗ δ ) = 0, 0 1 2 3 4 ¯ δ 5 6 7 8

introduction TG LE fixed n , m , f fixed n , m fixed n connected conclusion Laplacian Energy of Threshold Graphs � i − ¯ � i − ¯ � (¯ LE ( Th ( d ∗ )) = | d ∗ ( d ∗ δ − d ∗ δ | = δ ) + i ) i ∈ V i > ¯ i ≤ ¯ d ∗ δ d ∗ δ i − ¯ i − ¯ δ (¯ Note, � i ∈ V ( d ∗ δ ) = 0, thus � δ ( d ∗ δ ) = � δ − d ∗ i ) i > ¯ i ≤ ¯ d ∗ d ∗ 0 1 2 3 4 ¯ δ 5 6 7 8

introduction TG LE fixed n , m , f fixed n , m fixed n connected conclusion Laplacian Energy of Threshold Graphs � i − ¯ � i − ¯ � (¯ LE ( Th ( d ∗ )) = | d ∗ ( d ∗ δ − d ∗ δ | = δ ) + i ) i ∈ V i > ¯ i ≤ ¯ d ∗ δ d ∗ δ i − ¯ i − ¯ δ (¯ Note, � i ∈ V ( d ∗ δ ) = 0, thus � δ ( d ∗ δ ) = � δ − d ∗ i ) and i > ¯ i ≤ ¯ d ∗ d ∗ � i − ¯ � (¯ LE ( Th ( d ∗ )) = 2 ( d ∗ δ − d ∗ δ ) = 2 i ) i ≥ ¯ i ≤ ¯ d ∗ δ d ∗ δ 0 1 2 3 4 ¯ δ 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: 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 = = 5 5 5 6 6 6 7 7 7 8 8 8 arbitrary lexmin degree 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: 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 = = 5 5 5 6 6 6 7 7 7 8 8 8 arbitrary lexmin degree lexmax degree If ¯ δ < f , lexmin will be an optimal arrangement: 0 0 1 1 2 2 3 3 4 4 ≤ 5 5 6 6 7 7 8 8 arbitrary 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: 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 = = 5 5 5 6 6 6 7 7 7 8 8 8 arbitrary lexmin degree lexmax degree If ¯ δ < f , lexmin will be an optimal arrangement: 0 0 1 1 2 2 3 3 4 4 ≤ 5 5 6 6 7 7 8 8 arbitrary 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 of 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) ≤ 2 m and f ( f + 1) + 2( n − 1 − f ) f ≥ 2 m resulting in upper and lower bounds on f , � � � � � � n − 1 n 2 − n + 1 − 1 2 m + 1 =: ¯ f := 2 − 4 − 2 m ≤ f ≤ 2 + f . 4 Example for n = 7, m = 11 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 lexmin for ¯ lexmax for f = 2 lexmin for f = 3 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 .