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.