Spectral radius and average 2-degree sequence of a graph

Transcription

1 Spectral radius and average 2-degree sequence of a graph Speaker : Yu-pei Huang Advisor : Chih-wen Weng Department of Applied Mathematics, National Chiao Tung University 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 1 / 18

2 Average 2-degree sequence Let Γ=(X, R) denote a finite undirected, connected graph without loops or multiple edges with vertex set X, edge set R The spectral radius ρ(γ) of Γ is the largest eigenvalue of its adjacency matrix 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 2 / 18

3 Average 2-degree sequence Let Γ=(X, R) denote a finite undirected, connected graph without loops or multiple edges with vertex set X, edge set R The spectral radius ρ(γ) of Γ is the largest eigenvalue of its adjacency matrix For x X, we define the average 2-degree M x := y x d y/d x, where d x is the degree of x Label the vertices of Γ by 1, 2,, n such that M 1 M 2 M n 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 2 / 18

4 Average 2-degree sequence Let Γ=(X, R) denote a finite undirected, connected graph without loops or multiple edges with vertex set X, edge set R The spectral radius ρ(γ) of Γ is the largest eigenvalue of its adjacency matrix For x X, we define the average 2-degree M x := y x d y/d x, where d x is the degree of x Label the vertices of Γ by 1, 2,, n such that M 1 M 2 M n A graph of order n with identical average 2-degree (ie M 1 = M 2 = = M n ) is called pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 2 / 18

5 Non-regular pseudo-regular graphs 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 3 / 18

6 Non-regular pseudo-regular graphs Surrounded surrounded by 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 4 / 18

7 Perron-Frobenius Theorem The following theorem is a fundamental result on the study of Matrix Theory It is referred to as Perron-Frobenius Theorem Theorem If B is a nonnegative irreducible n n matrix with largest eigenvalue ρ(b) and row-sums r 1, r 2,, r n, then ρ(b) max 1 i n r i with equality if and only if the row-sums of B are all equal 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 5 / 18

8 An Application of Perron-Frobenius Theorem A = 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 6 / 18

9 An Application of Perron-Frobenius Theorem A = r 1 = 6 r 2 = 4 r 3 = 3 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 6 / 18

10 An Application of Perron-Frobenius Theorem A = r 1 = 6 r 2 = 4 r 3 = 3 = ρ(a) 6 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 6 / 18

11 An Application of Perron-Frobenius Theorem A = r 1 = 6 r 2 = 4 r 3 = 3 = ρ(a) 6 Consider A T, ρ(a) 5 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 6 / 18

12 An Application of Perron-Frobenius Theorem A = r 1 = 6 r 2 = 4 r 3 = 3 = ρ(a) 6 Consider A T, ρ(a) 5 In fact, e(a) = 1, 2 6, 2 + 6, ρ(a) = 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 6 / 18

13 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 7 / 18

14 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular Proof 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 7 / 18

15 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular Proof Let A be the adjacency matrix of a connected graph Γ Setting B = (b ij ) = U 1 AU, where U = diag (d 1, d 2,, d n ) Let r l be the l-th row sum of B 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 7 / 18

16 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular Proof Let A be the adjacency matrix of a connected graph Γ Setting B = (b ij ) = U 1 AU, where U = diag (d 1, d 2,, d n ) Let r l be the l-th row sum of B Then b ij = d j a ij /d i and r l = M l 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 7 / 18

17 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular Proof Let A be the adjacency matrix of a connected graph Γ Setting B = (b ij ) = U 1 AU, where U = diag (d 1, d 2,, d n ) Let r l be the l-th row sum of B Then b ij = d j a ij /d i and r l = M l Since A and B are similar, applying Perron-Frobenius Theorem to B, we have ρ(γ) M 1 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 7 / 18

18 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 8 / 18

19 Upper bounds of spectral radii Theorem Let Γ be a connected graph Then ρ(γ) M 1 with equality if and only if Γ is pseudo-regular Theorem (Chen, Pan and Zhang, 2011) Let Γ be a connected graph Let a := max {d i /d j 1 i, j n} Then ρ(γ) M 2 a + (M 2 + a) 2 + 4a(M 1 M 2 ), 2 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 8 / 18

20 Main result The following Theorem is our main result which is a generalization of the previous theorem Theorem For any b max {d i /d j i j} and 1 l n, ρ(γ) M l b + (M l + b) 2 + 4b l 1 i=1 (M i M l ) := ϕ l, 2 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 9 / 18

21 Main result The following Theorem is our main result which is a generalization of the previous theorem Theorem For any b max {d i /d j i j} and 1 l n, ρ(γ) M l b + (M l + b) 2 + 4b l 1 i=1 (M i M l ) := ϕ l, 2 with equality if and only if Γ is pseudo-regular Proof For each 1 i l 1, let x i 1 be a variable to be determined later Let U = diag(d 1 x 1,, d l 1 x l 1, d l,, d n ) be a diagonal matrix of size n n Consider the matrix B = U 1 AU Note that A and B have the same eigenvalues Let r 1, r 2,, r n be the row-sums of B 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 9 / 18

22 Proof (continue) Then for 1 i l 1, let we have x i = 1 + M i M l ϕ l + b 1 r i = l 1 k=1 1 d i x i a ik d k x k + n k=l 1 d i x i a ik d k 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 10 / 18

23 Proof (continue) Then for 1 i l 1, let we have r i = l 1 k=1 x i = 1 + M i M l ϕ l + b 1 1 d i x i a ik d k x k + n k=l = 1 l 1 d k (x k 1)a ik + 1 x i d i x i k=1 1 d i x i a ik d k n k=1 a ik d k d i 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 10 / 18

24 Proof (continue) Then for 1 i l 1, let we have x i = 1 + M i M l ϕ l + b 1 r i = l 1 k=1 1 d i x i a ik d k x k + n k=l 1 d i x i a ik d k = 1 l 1 d k (x k 1)a ik + 1 n d k a ik x i d i x i d i k=1 k=1 b l 1 x k (l 2) + 1 M i x i x i k=1,k i 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 10 / 18

25 Proof (continue) Then for 1 i l 1, let we have x i = 1 + M i M l ϕ l + b 1 r i = l 1 k=1 1 d i x i a ik d k x k + n k=l 1 d i x i a ik d k = 1 l 1 d k (x k 1)a ik + 1 n d k a ik x i d i x i d i k=1 k=1 b l 1 x k (l 2) + 1 M i x i x i = ϕ l k=1,k i 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 10 / 18

26 Proof (continue) Similarly for l j n we have r j = l 1 d k x k a jk + d j k=1 The first part of this theorem follows = n k=l a jk d k d j l 1 d k n d k (x k 1)a jk + a jk d j d j k=1 k=1 ( l 1 ) b x k (l 1) + M l = ϕ l k=1 Hence by Perron-Frobenius Theorem, ρ(γ) = ρ(b) max 1 i n {r i} ϕ l 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 11 / 18

27 Proof (equality part) To prove the sufficient condition, suppose M 1 = M 2 = = M n Then ρ(γ) = M 1 = ϕ l Hence the equality follows To prove the necessary condition, suppose ρ(γ) = ϕ l Hence r 1 = r 2 = = r n = ϕ l, and the equalities related hold In particular, b = a ik d k d i for any 1 i n and 1 k l 1 with k i and x k 1 > 0, and M l = M n We separate the condition into three cases: 1 M 1 = M l : Clearly M 1 = M n 2 M t 1 > M t = M l for some 3 t l: Then x k > 1 for 1 k t 1 d Hence b = a 2 d 12 d 1 = a 1 21 d 2 = 1, and d i = n 1 for all i = 1, 2,, n 3 M 1 > M 2 = M l : Then x 1 > 1 Hence b = a i1 d 1 /d i for 2 i n Hence d 1 = n 1 and d 2 = d 3 = = d n = (n 1)/b Then (n 1)/b = M 1 > M 2 = M n = (n 1)/b 1 + b 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 12 / 18

28 Comparing two theorems Theorem (Chen, Pan and Zhang, 2011) Let a := max {d i /d j 1 i, j n} Then ρ(γ) M 2 a + (M 2 + a) 2 + 4a(M 1 M 2 ), 2 with equality if and only if Γ is pseudo-regular Theorem For any b max {d i /d j i j} and 1 l n, ρ(γ) M l b + (M l + b) 2 + 4b l 1 i=1 (M i M l ), 2 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 13 / 18

29 Comparing two theorems Given a decreasing sequence M 1 M 2 M n of positive integers, consider the functions for x [1, ) ϕ l (x) = M l x + (M l + x) 2 + 4x l 1 i=1 (M i M l ) 2 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 14 / 18

30 Comparing two theorems Given a decreasing sequence M 1 M 2 M n of positive integers, consider the functions ϕ l (x) = M l x + (M l + x) 2 + 4x l 1 i=1 (M i M l ) 2 for x [1, ) Proposition For any 1 l n, ϕ l (x) is increasing on [1, ) 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 14 / 18

31 Comparing two theorems Given a decreasing sequence M 1 M 2 M n of positive integers, consider the functions ϕ l (x) = M l x + (M l + x) 2 + 4x l 1 i=1 (M i M l ) 2 for x [1, ) Proposition For any 1 l n, ϕ l (x) is increasing on [1, ) Proposition Suppose M s > M s+1 for some 1 s n 1, and let the symbol denote > or = Then ϕ s (x) ϕ s+1 (x) iff s M i xs(s 1) 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 14 / 18 i=1

32 Comparing two theorems Theorem (Chen, Pan and Zhang, 2011) Let a := max {d i /d j 1 i, j n} Then ρ(γ) M 2 a + (M 2 + a) 2 + 4a(M 1 M 2 ), 2 with equality if and only if Γ is pseudo-regular Theorem For any b max {d i /d j i j} and 1 l n, ρ(γ) M l b + (M l + b) 2 + 4b l 1 i=1 (M i M l ), 2 with equality if and only if Γ is pseudo-regular 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 15 / 18

33 A graph with ϕ 2 > ϕ 3 Example In the following graph, M 1 = M 2 = 4, M 3 = 7/2, b = 4/3, ϕ 1 = ϕ 2 = 4, ϕ and ρ(γ) = 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 16 / 18

34 Another graph with ϕ 2 > ϕ 3 Example In the following graph, M 1 = 14/3, M 2 = 4, M 3 = 35, a = 5, b = 4, ϕ 1 = 14/3, ϕ 2 (a) 4356, ϕ 2 (b) 4320, ϕ , and ρ(γ) / 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 17 / 18

35 Thank you for your attention! 黃喻培 (Dep of A Math, NCTU) Spectral radius and average 2-degree sequence of a graph 18 / 18