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Optimization of Eigenvalue Bounds for the Independence and Chromatic Number of Graph Powers

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Optimization of Eigenvalue Bounds for the Independence and Chromatic Number of Graph Powers Optimization of eigenvalue bounds for the independence and chromatic number of graph powers Sjanne Zeijlemaker Joint work with A. Abiad, G. Coutinho, M.A. Fiol, B.D. Nogueira 01-02-2021 Eindhoven University of Technology Outline 1. Background 2. Spectral bounds: an overview 3. The spectrum of 퐺푘 and 퐺 are related 4. Optimization of inertial type bounds 5. Two ratio type bounds 6. Closing remarks 1 Background Eigenvalues Spectrum: 휆1 ≥ … ≥ 휆푛 3, 1, 1, 1, 1, 1, −2, −2, −2, −2 2 Eigenvalues 푚0 푚푑 Spectrum: {휃0 , … , 휃푑 } 31, 15, −24 2 Independence number Independence number 훼(퐺): size of the largest independent set of vertices in 퐺 3 Independence number Independence number 훼(퐺): size of the largest independent set of vertices in 퐺 3 Chromatic number Chromatic number 휒(퐺): minimum number of colors needed to color 퐺 4 Graph powers The 푘푡ℎ power of a graph 퐺 = (푉 , 퐸), denoted by 퐺푘, is formed by connecting two vertices if they are at distance at most 푘 푘 = 1 5 Graph powers The 푘푡ℎ power of a graph 퐺 = (푉 , 퐸), denoted by 퐺푘, is formed by connecting two vertices if they are at distance at most 푘 푘 = 2 5 Graph powers The 푘푡ℎ power of a graph 퐺 = (푉 , 퐸), denoted by 퐺푘, is formed by connecting two vertices if they are at distance at most 푘 푘 = 3 5 푘-independence number 푘-independence number 훼푘(퐺): maximum size of a set of vertices at pairwise distance greater than 푘 훼2 6 푘-independence number 푘-independence number 훼푘(퐺): maximum size of a set of vertices at pairwise distance greater than 푘 훼2 6 푘-independence number 푘-independence number 훼푘(퐺): maximum size of a set of vertices at pairwise distance greater than 푘 훼2 푘 Note: 훼푘(퐺) = 훼(퐺 ) 6 푘-chromatic number 푘 푘-chromatic number 휒푘(퐺): 휒푘(퐺) = 휒(퐺 ) 휒2 7 푘-chromatic number 푘 푘-chromatic number 휒푘(퐺): 휒푘(퐺) = 휒(퐺 ) Upper bounds on 훼푘 give lower bounds on 휒푘 and vice versa: |푉 (퐺)| 휒(퐺) ≥ 훼(퐺) 7 Our goal Extend and optimize eigenvalue upper bounds for the independence number to 훼푘. 8 Eigenvalues can be computed in polynomial time. • Could we apply eigenvalue bounds on 훼 to 퐺푘? No, in general the spectrum of 퐺푘 cannot be derived from 퐺 and vice versa. ⇒ We want a bound that only depends on the spectrum of 퐺. Motivation • (Kong and Zhao 1993) Computing 훼푘 and 휒푘 is NP-complete (Kong and Zhao 2000) Even for regular bipartite graphs, the problem remains NP-complete if 푘 = 2, 3, 4 9 • Could we apply eigenvalue bounds on 훼 to 퐺푘? No, in general the spectrum of 퐺푘 cannot be derived from 퐺 and vice versa. ⇒ We want a bound that only depends on the spectrum of 퐺. Motivation • (Kong and Zhao 1993) Computing 훼푘 and 휒푘 is NP-complete (Kong and Zhao 2000) Even for regular bipartite graphs, the problem remains NP-complete if 푘 = 2, 3, 4 Eigenvalues can be computed in polynomial time. 9 No, in general the spectrum of 퐺푘 cannot be derived from 퐺 and vice versa. ⇒ We want a bound that only depends on the spectrum of 퐺. Motivation • (Kong and Zhao 1993) Computing 훼푘 and 휒푘 is NP-complete (Kong and Zhao 2000) Even for regular bipartite graphs, the problem remains NP-complete if 푘 = 2, 3, 4 Eigenvalues can be computed in polynomial time. • Could we apply eigenvalue bounds on 훼 to 퐺푘? 9 ⇒ We want a bound that only depends on the spectrum of 퐺. Motivation • (Kong and Zhao 1993) Computing 훼푘 and 휒푘 is NP-complete (Kong and Zhao 2000) Even for regular bipartite graphs, the problem remains NP-complete if 푘 = 2, 3, 4 Eigenvalues can be computed in polynomial time. • Could we apply eigenvalue bounds on 훼 to 퐺푘? No, in general the spectrum of 퐺푘 cannot be derived from 퐺 and vice versa. 9 Motivation • (Kong and Zhao 1993) Computing 훼푘 and 휒푘 is NP-complete (Kong and Zhao 2000) Even for regular bipartite graphs, the problem remains NP-complete if 푘 = 2, 3, 4 Eigenvalues can be computed in polynomial time. • Could we apply eigenvalue bounds on 훼 to 퐺푘? No, in general the spectrum of 퐺푘 cannot be derived from 퐺 and vice versa. ⇒ We want a bound that only depends on the spectrum of 퐺. 9 Applications • Coding theory: codes relate to 푘-independent sets in Hamming graphs. 111 110 101 011 100 010 001 000 10 Applications • Coding theory: codes relate to 푘-independent sets in Hamming graphs. • Quantum independence number 훼푞(퐺) (Roberson and Mancinska 2016). Not known whether 훼푘푞(퐺) is generally computable. Some of our bounds on 훼푘(퐺) also upper bound 훼푘푞(퐺) (Wocjan, Elphick and Abiad 2019). • Other graph parameters: the 푘-independence number can be used to get tight lower bounds for the average distance (Firby and Haviland 1997). 10 Main tool: interlacing Let 푚 < 푛. Sequences 휆1 ≥ … ≥ 휆푛 and 휇1 ≥ … ≥ 휇푚 interlace if 휆푖 ≥ 휇푖 ≥ 휆푛−푚+푖 (1 ≤ 푖 ≤ 푚) 11 Main tool: interlacing Let 푚 < 푛. Sequences 휆1 ≥ … ≥ 휆푛 and 휇1 ≥ … ≥ 휇푚 interlace if 휆푖 ≥ 휇푖 ≥ 휆푛−푚+푖 (1 ≤ 푖 ≤ 푚) 휇1 휇2 휇3 휆1 휆2 휆3 휆4 휆5 11 Main tool: interlacing Let 푚 < 푛. Sequences 휆1 ≥ … ≥ 휆푛 and 휇1 ≥ … ≥ 휇푚 interlace if 휆푖 ≥ 휇푖 ≥ 휆푛−푚+푖 (1 ≤ 푖 ≤ 푚) 휇1 휇2 휇3 휇4 휆1 휆2 휆3 휆4 휆5 푚 = 푛 − 1 11 Eigenvalue interlacing Symmetric matrix 퐴: 휆1 ≥ … ≥ 휆푛 Matrix 퐵: 휇1 ≥ … ≥ 휇푚 12 Eigenvalue interlacing Symmetric matrix 퐴: 휆1 ≥ … ≥ 휆푛 Matrix 퐵: 휇1 ≥ … ≥ 휇푚 Cauchy interlacing: if 퐵 is a principal submatrix of 퐴, then the eigenvalues of 퐵 interlace those of 퐴. 12 Eigenvalue interlacing Symmetric matrix 퐴: 휆1 ≥ … ≥ 휆푛 Matrix 퐵: 휇1 ≥ … ≥ 휇푚 (Haemers 1995) if 퐵 is the quotient matrix of a partition of 퐴, then the eigenvalues of 퐵 interlace the eigenvalues of 퐴. um w s 퐵 average ro 퐴 12 Spectral bounds: an overview and 푛 휒(퐺) ≥ . min{푁 +, 푁 −} Classic bounds Inertia bound (Cvetković 1972) If 퐺 is a graph with eigenvalues 휆1, … , 휆푛, then 훼(퐺) ≤ min{|푖 ∶ 휆푖 ≥ 0|, |푖 ∶ 휆푖 ≤ 0|} = min{푁 +, 푁 −} 13 Classic bounds Inertia bound (Cvetković 1972) If 퐺 is a graph with eigenvalues 휆1, … , 휆푛, then 훼(퐺) ≤ min{|푖 ∶ 휆푖 ≥ 0|, |푖 ∶ 휆푖 ≤ 0|} = min{푁 +, 푁 −} and 푛 휒(퐺) ≥ . min{푁 +, 푁 −} 13 Classic bounds Ratio bound (Hoffman 1970) If 퐺 is regular with eigenvalues 휆1, … , 휆푛, then −휆 훼(퐺) ≤ 푛 푛 휆1 − 휆푛 and if an independent set 퐶 meets this bound then every vertex not in 퐶 is adjacent to precisely −휆푛 vertices of 퐶. (Lovász 1979) The Lovász theta number 휗(퐺) is a lower bound for the Hoffman bound. 14 Classic bounds: application Theorem (Erdős, Ko and Rado 1961) Let 푛 ≥ 2푘 and let ℱ be a collection of 푘-subsets of [푛] such that any 푛−1 two sets intersect. Then |ℱ| ≤ (푘−1). [푛] The Kneser graph 퐾퐺푛,푘 has vertex set ( 푘 ) with edges defined by void intersection. 13 45 25 24 34 15 23 12 14 35 퐾퐺5,2 15 Classic bounds: application Theorem (Erdős, Ko and Rado 1961) Let 푛 ≥ 2푘 and let ℱ be a collection of 푘-subsets of [푛] such that any 푛−1 two sets intersect. Then |ℱ| ≤ (푘−1). [푛] The Kneser graph 퐾퐺푛,푘 has vertex set ( 푘 ) with edges defined by void intersection. ⇕ Theorem (EKR) 푛−1 훼(퐾퐺푛,푘) ≤ (푘−1). This follows directly from the Hoffman bound. 15 Bounds on 훼푘 • (Firby and Haviland 1997) Upper bound for 훼푘 in a connected graph. • (Fiol 1997) Eigenvalue upper bound for regular graphs using alternating polynomials. • (Beis, Duckworth and Zito 2005) Upper bounds for 훼푘 in random 푟-regular graphs. • (O, Shi and Taoqiu 2019) Tight upper bounds in an 푛-vertex 푟-regular graph for every 푘 ≥ 2 and 푟 ≥ 3. • (Jou, Lin and Lin 2020) Tight upper bound for the 2-independence number of a tree. 16 Optimization and eigenvalue bounds Independence number: • (Delsarte 1973) LP bound on 훼 for distance-regular graphs. • (Lovász 1979) SDP bound 휗. 푘-independence number: Hoffman ratio bound Inertia bound ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ (Fiol 2019) ? LP with minor polynomials 17 What about general degree-푘 polynomials? (2) (Abiad, Coutinho and Fiol 2019) Extend previous bounds to 푝(휆푖) for some polynomial 푝 of degree 푘. Which polynomial gives the best bound for a specific graph? (3) (Abiad, Coutinho, Fiol, Nogueira and Z. 2020) Optimize previous bounds over 푝 ∈ ℝ푘[푥]. Line of work 푘 (1) (Abiad, Cioabă and Tait 2016) New bounds on 훼푘 in terms of 휆푖 . 18 (2) (Abiad, Coutinho and Fiol 2019) Extend previous bounds to 푝(휆푖) for some polynomial 푝 of degree 푘. Which polynomial gives the best bound for a specific graph? (3) (Abiad, Coutinho, Fiol, Nogueira and Z. 2020) Optimize previous bounds over 푝 ∈ ℝ푘[푥]. Line of work 푘 (1) (Abiad, Cioabă and Tait 2016) New bounds on 훼푘 in terms of 휆푖 . What about general degree-푘 polynomials? 18 Which polynomial gives the best bound for a specific graph? (3) (Abiad, Coutinho, Fiol, Nogueira and Z. 2020) Optimize previous bounds over 푝 ∈ ℝ푘[푥]. Line of work 푘 (1) (Abiad, Cioabă and Tait 2016) New bounds on 훼푘 in terms of 휆푖 . What about general degree-푘 polynomials? (2) (Abiad, Coutinho and Fiol 2019) Extend previous bounds to 푝(휆푖) for some polynomial 푝 of degree 푘.
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