Some results on the roots of the independence polynomial of graphs

Transcription

1 Some results on the roots of the independence polynomial of graphs Ferenc Bencs Central European University Algebraic and Extremal Graph Theory Conference Aug 9, 2017 Supervisor: Péter Csikvári

2 Denition, notation The independence polynomial of a graph G is I(G, x) = x A =, A F(G) where F(G) is the set of independent subsets of the vertices of G. E.g. 2 / 18

3 Denition, notation The independence polynomial of a graph G is I(G, x) = x A = i k (G)x k, A F(G) k 0 where F(G) is the set of independent subsets of the vertices of G. E.g. 2 / 18

4 Denition, notation The independence polynomial of a graph G is I(G, x) = x A = i k (G)x k, A F(G) k 0 where F(G) is the set of independent subsets of the vertices of G. E.g I(G, x) = 1 + 5x + 3x 2 2 / 18

5 Notation For a generater function f(x) = k 0 a kx k, we will use the following notation to denote the coecient of x k in f(x). [x k ]f(x) = a k 3 / 18

6 Unimodality A sequence (b k ) n k=0 R+ is 1. unimodal, if k {0,... n}, such that b 0 b 1 b k 1 b k b k+1 b n. 2. log-concave, if i {1,..., n 1} b 2 i b i 1b i+1 Lemma (Newton) If p(x) = n i=0 b ix i has only real zeros, then the sequence (b k ) n k=0 is log-concave, therefore unimodal. 4 / 18

7 Unimodality on the independent subsets of the graphs Question: Are the coecients of I(G, x) form an unimodal sequence, if G is...? connected bipartite (Levit, Mandrescu) tree (Alavi et al.) line graph claw-free graph Answer: No No (Bhattacharyya, Kahn) Open Yes Yes 5 / 18

8 Unimodality on the independent subsets of the graphs Question: Are the coecients of I(G, x) form an unimodal sequence, if G is...? connected bipartite (Levit, Mandrescu) tree (Alavi et al.) line graph claw-free graph Answer: No No (Bhattacharyya, Kahn) Open Yes, moreover real-rooted (Heilmann, Lieb) Yes, moreover real-rooted (Chudnovsky, Seymour) (claw-free=graph without induced K 1,3.) 5 / 18

9 Unimodality on the independent subsets of the graphs Question: Are the coecients of I(G, x) form an unimodal sequence, if G is...? connected bipartite (Levit, Mandrescu) tree (Alavi et al.) line graph claw-free graph Answer: No No (Bhattacharyya, Kahn) Open Yes, moreover real-rooted (Heilmann, Lieb) Yes, moreover real-rooted (Chudnovsky, Seymour) 5 / 18

10 Unimodality on the independent subsets of the graphs Question: Are the coecients of I(G, x) form an unimodal sequence, if G is...? connected bipartite (Levit, Mandrescu) tree (Alavi et al.) line graph claw-free graph Answer: No No (Bhattacharyya, Kahn) Open Yes, moreover real-rooted (Heilmann, Lieb) Yes, moreover real-rooted (Chudnovsky, Seymour) Question(Galvin, Hilyard): Which trees have independence polynomial with only real zeros? 5 / 18

11 Stable-path tree Let u be a xed vertex of G, and choose a total ordering on V (G). Then the rooted tree (T (G, u), ū) dened as follows: Let N(u) = {u 1 u d } and G i = G[V (G) \ {u, u 1, u 2,..., u i 1 }] 6 / 18

12 Stable-path tree Let u be a xed vertex of G, and choose a total ordering on V (G). Then the rooted tree (T (G, u), ū) dened as follows: Let N(u) = {u 1 u d } and G i = G[V (G) \ {u, u 1, u 2,..., u i 1 }] ū 1 T (G 1, u 1 ) ū 2 T (G 2, u 2 ). ū d T (G d, u d )

13 Stable-path tree Let u be a xed vertex of G, and choose a total ordering on V (G). Then the rooted tree (T (G, u), ū) dened as follows: Let N(u) = {u 1 u d } and G i = G[V (G) \ {u, u 1, u 2,..., u i 1 }] ū 1 T (G 1, u 1 ) ū ū 2 ū d T (G 2, u 2 ). T (G d, u d )

14 Stable-path tree Let u be a xed vertex of G, and choose a total ordering on V (G). Then the rooted tree (T (G, u), ū) dened as follows: Let N(u) = {u 1 u d } and G i = G[V (G) \ {u, u 1, u 2,..., u i 1 }] ū 1 T (G 1, u 1 ) ū ū 2 ū d T (G 2, u 2 ). T (G d, u d ) 6 / 18

15 Example / 18

16 Example

17 Example

18 Example / 18

19 Example / 18

20 Example -cont / 18

21 Properties of the stable-path tree Theorem (Scott, Sokal, Weitz) Let G be a graph and u V (G) be xed. Then if T = T (G, u), then I(G u, x) I(G, x) = I(T u, x). I(T, x) 9 / 18

22 Properties of the stable-path tree Theorem (Scott, Sokal, Weitz) Let G be a graph and u V (G) be xed. Then if T = T (G, u), then I(G u, x) I(G, x) = I(T u, x). I(T, x) Corollary 1. There exists a subtree F in T such that I(G, x) = I(T, x) I(T F, x). 9 / 18

23 Properties of the stable-path tree Theorem (Scott, Sokal, Weitz) Let G be a graph and u V (G) be xed. Then if T = T (G, u), then I(G u, x) I(G, x) = I(T u, x). I(T, x) Corollary 1. There exists a subtree F in T such that I(G, x) = I(T, x) I(T F, x). 2. There exists a sequence of induced subgraphs G 1,..., G k of G, such that I(T, x) = I(G, x)i(g 1, x)... I(G k, x). 9 / 18

24 Real-rooted independence polynomials Theorem (Chudnovsky, Seymour) Any zero of the independence polynomial of a claw-free graph is real. (claw-free=graph without induced K 1,3.) 10 / 18

25 Real-rooted independence polynomials Theorem (Chudnovsky, Seymour) Any zero of the independence polynomial of a claw-free graph is real. (claw-free=graph without induced K 1,3.) Theorem Let G be a claw-free graph and u V (G). Then I(T (G, u), x) is real-rooted. Moreover I(G, x) divides I(T (G, u), x). 10 / 18

26 Caterpillar H n The nth caterpillar (H n) is the following tree on 3n vertices: / 18

27 Caterpillar H n The nth caterpillar (H n) is the following tree on 3n vertices:... Proposition: It has real-rooted independence polynomial.(wang, Zhu) 11 / 18

28 Caterpillar H n The nth caterpillar (H n) is the following tree on 3n vertices:... Proposition: It has real-rooted independence polynomial.(wang, Zhu) For a tree T, call a claw-free graph witness, if there is an ordering of its vertices, such that the resulting stable-path tree is isomorphic to T. 11 / 18

29 Caterpillar H n The nth caterpillar (H n) is the following tree on 3n vertices:... Proposition: It has real-rooted independence polynomial.(wang, Zhu) For a tree T, call a claw-free graph witness, if there is an ordering of its vertices, such that the resulting stable-path tree is isomorphic to T. The witness is the following graph n + 2 n + 3 2n n 1 n n / 18

30 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), 12 / 18

31 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 1

32 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 1 12 / 18

33 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 1 12 / 18

34 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 1 12 / 18

35 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 1 r 3 12 / 18

36 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 1 r 3 12 / 18

37 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 4 r 1 r 3 12 / 18

38 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 4 r 1 r 3 Conjecture: It has real-rooted independence polynomial. (Galvin, Hilyard) 12 / 18

39 Fibonaci tree F n The Fibonaci trees are dened recursively (denition by S. Wagner), r 0 r 2 r 4 r 1 r 3 Conjecture: It has real-rooted independence polynomial. (Galvin, Hilyard) The witness is the following graph n 3 n 2 n 1 12 / 18

40 The M n graph family M n when n is even. M n when n is odd. 13 / 18

41 The M n graph family M n when n is even. M n when n is odd. Proposition: It has real-rooted independence polynomial. (Wang, Zhu) 13 / 18

42 The M n graph family M n when n is even. M n when n is odd. Proposition: It has real-rooted independence polynomial. (Wang, Zhu) As it turns out with a good ordering on the vertices the corresponding stable-path tree will be the nth caterpillar. 13 / 18

43 The M n graph family M n when n is even. M n when n is odd. Proposition: It has real-rooted independence polynomial. (Wang, Zhu) As it turns out with a good ordering on the vertices the corresponding stable-path tree will be the nth caterpillar. So I(M n, x) I(H n, x), and we already seen that I(H n, x) has only real zeros, therefore I(M n, x) has only real zeros. 13 / 18

44 Other real-rooted families Trees: Graphs: 14 / 18

45 Other real-rooted families Trees: Centipede (Zhu) Graphs: / 18

46 Other real-rooted families Trees: Graphs: Centipede (Zhu) k-ary analogue of the Fibbonaci trees (denition by S. Wagner) / 18

47 Other real-rooted families Trees: Graphs: Centipede (Zhu) k-ary analogue of the Fibbonaci trees (denition by S. Wagner) Sunlet graph (Wang, Zhu) / 18

48 Other real-rooted families Trees: Graphs: Centipede (Zhu) k-ary analogue of the Fibbonaci trees (denition by S. Wagner) Sunlet graph (Wang, Zhu) Ladder graph (Zhu, Lu) / 18

49 Other real-rooted families Trees: Graphs: Centipede (Zhu) k-ary analogue of the Fibbonaci trees (denition by S. Wagner) Sunlet graph (Wang, Zhu) Ladder graph (Zhu, Lu) Polyphenil ortho-chain (Alikhani, Jafari) / 18

50 Are they the answer? Is there a tree with real-rooted independence polynomial, such that it is not a stable-path tree of other then itself? 15 / 18

51 Are they the answer? Is there a tree with real-rooted independence polynomial, such that it is not a stable-path tree of other then itself? Yes. E.g.: I(T, x) = (1+x)(1+8x+20x 2 +16x 3 +x 4 ) 15 / 18

52 Dictionary Finite graphs Innite (rooted) graphs 16 / 18

53 Dictionary Finite graphs zeros of a polynomial Innite (rooted) graphs 16 / 18

54 Dictionary Finite graphs zeros of a polynomial Innite (rooted) graphs measure on the complex plane 16 / 18

55 Dictionary Finite graphs zeros of a polynomial all zeros are real Innite (rooted) graphs measure on the complex plane 16 / 18

56 Dictionary Finite graphs zeros of a polynomial all zeros are real Innite (rooted) graphs measure on the complex plane measure supported on the real line 16 / 18

57 Dictionary Finite graphs zeros of a polynomial all zeros are real Innite (rooted) graphs measure on the complex plane measure supported on the real line The main ingredient which enables us to move between the two worlds, is the localization. For any graph G, the coecient [x k I(G u, x) ] I(G, x) depends only on the k 1-neighborhood of u in G, 16 / 18

58 Dictionary Finite graphs zeros of a polynomial all zeros are real Innite (rooted) graphs measure on the complex plane measure supported on the real line The main ingredient which enables us to move between the two worlds, is the localization. For any graph G, the coecient [x k I(G u, x) ] I(G, x) depends only on the k 1-neighborhood of u in G, moreover it is a positive integer. 16 / 18

59 Innite binary tree Theorem For the rooted binary tree (or 3-regular tree) there exists a measure µ on the real line, such that R x k dµ = [x k I(T r, x) ] I(T, x) 17 / 18

60 Innite binary tree Theorem For the rooted binary tree (or 3-regular tree) there exists a measure µ on the real line, such that R = 1 k x k dµ = [x k I(T r, x) ] I(T, x) ( 3k ) k 1 17 / 18

61 Innite binary tree Theorem For the rooted binary tree (or 3-regular tree) there exists a measure µ on the real line, such that R x k dµ = [x k I(T r, x) ] I(T, x) = k i=1 i ( 3k ) k k i 17 / 18

62 THANK YOU FOR YOUR ATTENTION! 18 / 18