Hermitian adjacency matrix of digraphs

Transcription

1 of digraphs Krystal Guo Bojan Mohar Department of Combinatorics & Optimization Univeristy of Waterloo Systems of Lines: Applications of Algebraic Combinatorics, August 12, 2015 K. Guo of digraphs

2 X A(X) = Spec(X) = {2, ( 1) (2) } K. Guo of digraphs

3 X A(X) = Spec(X) = {2, ( 1) (2) } Graph Adjacency matrix Algebraic properties K. Guo of digraphs

4 X A(X) = Spec(X) = {2, ( 1) (2) } Graph Adjacency matrix Algebraic properties K. Guo of digraphs

5 X A(X) = Spec(X) = {2, ( 1) (2) } Graph Adjacency matrix Algebraic properties K. Guo of digraphs

6 X A(X) = Spec(X) = {2, ( 1) (2) } Graph ma- Adjacency trix Algebraic properties K. Guo of digraphs

7 X A(X) = Spec(X) = {2, ( 1) (2) } Digraph ma- Adjacency trix Algebraic properties K. Guo of digraphs

8 X A(X) = Spec(X) = {2, ( 1) (2) } Digraph? Algebraic properties K. Guo of digraphs

9 u u v x y v u i x X v H(X) = x i 1 y y 1 K. Guo of digraphs

10 Background For graphs, H(G) = A(G). K. Guo of digraphs

11 Background For graphs, H(G) = A(G). For oriented graphs, H(D) = im(d), where M(D) is the skew symmetric matrix. Skew-symmetric tournament matrices have been well studied (skew two-graphs). K. Guo of digraphs

12 Background For graphs, H(G) = A(G). For oriented graphs, H(D) = im(d), where M(D) is the skew symmetric matrix. Skew-symmetric tournament matrices have been well studied (skew two-graphs). Skew-symmetric conference matrices have also been well studied (e.g. Delsarte, Goethals, Seidel 71) K. Guo of digraphs

13 Background For graphs, H(G) = A(G). For oriented graphs, H(D) = im(d), where M(D) is the skew symmetric matrix. Skew-symmetric tournament matrices have been well studied (skew two-graphs). Skew-symmetric conference matrices have also been well studied (e.g. Delsarte, Goethals, Seidel 71) More recently, the singular values of the skew-symmetric adjacency matrix of oriented graph has been studied (Adiga, Balakrishnan, and So in 2010; Cavers, Cioabă, et al in 2012; Hou and Lei in 2011). K. Guo of digraphs

14 Background For graphs, H(G) = A(G). For oriented graphs, H(D) = im(d), where M(D) is the skew symmetric matrix. Skew-symmetric tournament matrices have been well studied (skew two-graphs). Skew-symmetric conference matrices have also been well studied (e.g. Delsarte, Goethals, Seidel 71) More recently, the singular values of the skew-symmetric adjacency matrix of oriented graph has been studied (Adiga, Balakrishnan, and So in 2010; Cavers, Cioabă, et al in 2012; Hou and Lei in 2011). This matrix was independently defined by Liu and Li who used it to study energy of digraphs. K. Guo of digraphs

15 Why would we want to work with H(X)? K. Guo of digraphs

16 Why would we want to work with H(X)? H(X) is diagonalizable with real eigenvalues λ 1... λ n ; K. Guo of digraphs

17 Why would we want to work with H(X)? H(X) is diagonalizable with real eigenvalues λ 1... λ n ; Some graph eigenvalue techniques can be applied in this setting, so we hope to extend some spectral theorems to the class of digraphs. K. Guo of digraphs

18 Why would we want to work with H(X)? H(X) is diagonalizable with real eigenvalues λ 1... λ n ; Some graph eigenvalue techniques can be applied in this setting, so we hope to extend some spectral theorems to the class of digraphs. In particular, we may use eigenvalue interlacing. K. Guo of digraphs

19 Some strange things occur The largest eigenvalue in absolute value could in fact be negative; K. Guo of digraphs

20 Some strange things occur The largest eigenvalue in absolute value could in fact be negative; we can have λ n > λ 1. K. Guo of digraphs

21 Some strange things occur The largest eigenvalue in absolute value could in fact be negative; we can have λ n > λ 1. There does not appear to be a bound on the diameter of the digraph in terms of the number of distinct eigenvalues of the. K. Guo of digraphs

22 Some strange things occur The largest eigenvalue in absolute value could in fact be negative; we can have λ n > λ 1. There does not appear to be a bound on the diameter of the digraph in terms of the number of distinct eigenvalues of the. In fact, there is an infinite family of weakly connected digraphs whose number of distinct eigenvalues is constant, but whose diameter goes to infinity. K. Guo of digraphs

23 Spectral Radius Interlacing K 3 K 4 K. Guo of digraphs

24 Spectral Radius Interlacing K 3 K 4 These graphs have H-spectrum { 2, 1, 1} and { 3, 1, 1}, resp. K. Guo of digraphs

25 Spectral Radius Interlacing We consider the spectral radius ρ(x), which is the maximum absolute value amongst the H-eigenvalues of X. K. Guo of digraphs

26 Spectral Radius Interlacing We consider the spectral radius ρ(x), which is the maximum absolute value amongst the H-eigenvalues of X. Theorem For every digraph X we have Both inequalities are tight. λ 1 (X) ρ(x) 3λ 1 (X). K. Guo of digraphs

27 Spectral Radius Interlacing We denote the underlying graph of X by Γ(X). Theorem If X is a digraph, then ρ(x) (Γ(X)). When X is weakly connected, the equality holds if and only if Γ(X) is a (Γ(X))-regular graph and there exists a partition of V (X) into four parts as in figures below. V 1 V i V 1 V 1 V i V 1 V i V i K. Guo of digraphs

28 Spectral Radius Interlacing Theorem A digraph X has σ H (X) ( 3, 3 ) if and only if every component Y of X has Γ(Y ) isomorphic to a path of length at most 3 or to C 4. K. Guo of digraphs

29 Spectral Radius Interlacing Theorem A digraph X has σ H (X) ( 3, 3 ) if and only if every component Y of X has Γ(Y ) isomorphic to a path of length at most 3 or to C 4. Note: in the latter case we can exactly which digraphs Y is isomorphic to. K. Guo of digraphs

30 Spectral Radius Interlacing Eigenvalue interlacing {µ j } n 1 j=1 interlaces {λ j} n j=1 if: K. Guo of digraphs

31 Spectral Radius Interlacing Eigenvalue interlacing {µ j } n 1 j=1 interlaces {λ j} n j=1 if: λ n λ n 1 λ n 2 λ 2 λ 1... µ n 1 µ n 2 µ n 3 µ 1... K. Guo of digraphs

32 Spectral Radius Interlacing More generally, µ 1 µ m interlaces λ 1 λ n if λ j µ j λ n m+j. K. Guo of digraphs

33 Spectral Radius Interlacing More generally, µ 1 µ m interlaces λ 1 λ n if λ j µ j λ n m+j. If Y is an induced sub-digraph of X, then the eigenvalues of H(Y ) interlace those of H(X). K. Guo of digraphs

34 Spectral Radius Interlacing More generally, µ 1 µ m interlaces λ 1 λ n if λ j µ j λ n m+j. If Y is an induced sub-digraph of X, then the eigenvalues of H(Y ) interlace those of H(X). Interlacing is a powerful tool that is used in the undirected case to find eigenvalue bounds on many graph properties. K. Guo of digraphs

35 Spectral Radius Interlacing η + (X) = number of non-negative H-eigenvalues of X η (X) = number of non-positive H-eigenvalues of X. K. Guo of digraphs

36 Spectral Radius Interlacing η + (X) = number of non-negative H-eigenvalues of X η (X) = number of non-positive H-eigenvalues of X. Theorem If a digraph X contains a subset of m vertices, no two of which form a digon, then η + (X) m 2 and η (X) m 2. K. Guo of digraphs

37 Spectral Radius Interlacing η + (X) = number of non-negative H-eigenvalues of X η (X) = number of non-positive H-eigenvalues of X. Theorem If a digraph X contains a subset of m vertices, no two of which form a digon, then η + (X) m 2 and η (X) m 2. We can also generalize the Cvetković bound for the largest independent set to digraphs. K. Guo of digraphs

38 Spectral Radius Interlacing η + (X) = number of non-negative H-eigenvalues of X η (X) = number of non-positive H-eigenvalues of X. Theorem If a digraph X contains a subset of m vertices, no two of which form a digon, then η + (X) m 2 and η (X) m 2. We can also generalize the Cvetković bound for the largest independent set to digraphs. Theorem If X has an independent set of size α, then η + (X) α and η (X) α. K. Guo of digraphs

39 Spectral Radius Interlacing Theorem (Gregory, Kirkland, Shader 93) If X is an oriented graph of order n, then ( π ) λ 1 (H(X)) cot. 2n Equality holds if and only if X is switching-equivalent to T n, the transitive tournament of order n. K. Guo of digraphs

40 Spectral Radius Interlacing Theorem (Gregory, Kirkland, Shader 93) If X is an oriented graph of order n, then ( π ) λ 1 (H(X)) cot. 2n Equality holds if and only if X is switching-equivalent to T n, the transitive tournament of order n. Corollary If X is a digraph with an induced subdigraph that is switching equivalent to T m, then λ 1 (H(X)) cot ( π 2m). K. Guo of digraphs

41 Spectral Radius Interlacing Open problems Many problems concerning cospectrality of digraphs. K. Guo of digraphs

42 Spectral Radius Interlacing Open problems Many problems concerning cospectrality of digraphs. Bound on dichromatic number. K. Guo of digraphs

43 Spectral Radius Interlacing Open problems Many problems concerning cospectrality of digraphs. Bound on dichromatic number. Other analogues of undirected spectral bounds. K. Guo of digraphs

44 Spectral Radius Interlacing Thanks! This research is partially supported by NSERC. [?] K. Guo of digraphs