Spectra of digraph transformations Aiping Deng Donghua University, - - PowerPoint PPT Presentation

Spectra of digraph transformations

Spectra of digraph transformations

Aiping Deng

Donghua University, University of Wisconsin

Joint work with Alexander Kelmans

University of Puerto Rico, Rutgers University

June 2-5, 2014, Villanova

Spectra of digraph transformations

In this talk we introduce certain operations on digraphs depending

• n parameters x, y, z ∈ {0, 1, +, −}. The digraphs produced by

these operations are called the xyz-transformations of the original digraph. We establish the relationship between the adjacency/Laplacian spectra of certain kinds of digraphs and their xyz-transformations. Furthermore we present some pairs of adjacency/Laplacian cospectral non-isomorphic transformations of digraphs. We also give more constructions on adjacency/Laplacian cospectral non-isomorphic digraphs.

Spectra of digraph transformations

Contents

1 Definitions and notations

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D 3 Spectra of Dxyz

• for regular digraph D with possible loops

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D 3 Spectra of Dxyz

• for regular digraph D with possible loops

4 Spectra of D[xyz] for quasi-regular digraph D

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D 3 Spectra of Dxyz

• for regular digraph D with possible loops

4 Spectra of D[xyz] for quasi-regular digraph D 5 Spectra of F xyz for digraph-function F

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D 3 Spectra of Dxyz

• for regular digraph D with possible loops

4 Spectra of D[xyz] for quasi-regular digraph D 5 Spectra of F xyz for digraph-function F 6 Isomorphic and cospectral non-isomorphic xyz-transformations

Spectra of digraph transformations

Contents

1 Definitions and notations 2 Spectra of Dxyz for simple regular digraph D 3 Spectra of Dxyz

• for regular digraph D with possible loops

4 Spectra of D[xyz] for quasi-regular digraph D 5 Spectra of F xyz for digraph-function F 6 Isomorphic and cospectral non-isomorphic xyz-transformations 7 References

Spectra of digraph transformations Definitions and notations

Definitions and notations

A simple digraph D = (V, E) is a digraph with vertex set V and arc set E ⊆ V × V \ {(v, v) : v ∈ V }. D is the set of simple digraphs. The simple complete digraph K(V ) is the digraph in D with vertex set V and arc set V × V \ {(v, v) : v ∈ V }. D◦ is the set of digraphs D with vertex set V and arc set E(D) ⊆ V × V . The complete digraph K◦(V ) is the digraph in D◦ with vertex set V and arc set V × V .

Spectra of digraph transformations Definitions and notations

Let D = (V, E) ∈ D. D0 is the digraph with V (D0) = V and with no arcs. D1 is the simple complete digraph K(V ). D+ = D. D− is the simple complement of D, with vertex set V and arc set K(V ) \ E. Let D = (V, E) ∈ D◦. D0

• is the digraph with V (D0
• ) = V and with no arcs.

D1

• is the complete digraph K◦(V ).

D+

• = D.

D−

• is the complement of D, with vertex set V and arc set

K◦(V ) \ E.

Spectra of digraph transformations Definitions and notations

If e = (u, v) ∈ E(D), then put t(e, D) = t(e) = u and h(e, D) = h(e) = v. Dℓ is the line digraph of D, with V (Dℓ) = E(D) and E(Dℓ) = {(p, q) : p, q ∈ E(D) and h(p, D) = t(q, D)}. T(D) (resp. T cb(D)) is the digraph with vertex set V ∪ E and such that (v, e) is an arc in T(D) (resp. in T cb(D)) if and

• nly if v ∈ V, e ∈ E and vertex v = t(e) (resp. v = t(e)) in D.

H(D) (resp. Hcb(D)) is the digraph with vertex set V ∪ E and such that (e, v) is an arc in H(D) (resp. in Hcb(D)) if and only if v ∈ V, e ∈ E and vertex v = h(e) (resp. v = h(e)) in D.

Spectra of digraph transformations Definitions and notations

D T(D) Hcb(D) Tcb(D) H(D)

Figure: Digraphs T(D), H(D), T cb(D), and Hcb(D) of a dipath D.

Spectra of digraph transformations Definitions and notations

Definition Given a simple digraph D and three variables x, y, z ∈ {0, 1, +, −}, the xyz-transformation Dxyz of D is the digraph such that Dxy0 = Dx ∪ (Dl)y and Dxyz = Dxy0 ∪ W, where W = T(D) ∪ H(D) if z = +, W = T cb(D) ∪ Hcb(D) if z = −, and W is the union of the complete (V, E)-bipartite and the complete (E, V )-bipartite digraphs if z = 1.

Spectra of digraph transformations Definitions and notations

D D00+ D-++ D10+ D11- D-1- D+-- D01- D+0+ D+++ D---

Figure: Some xyz-transformations of the dipath D.

Spectra of digraph transformations Definitions and notations

Definition Given a digraph D ∈ D◦ and three variables x, y, z ∈ {0, 1, +, −}, the xyz-transformation Dxyz

• f D is the digraph such that

Dxy0

• = Dx
• ∪ (Dl)y and Dxyz
• = Dxy0
• ∪ W, where

W = T(D) ∪ H(D) if z = +, W = T cb(D) ∪ Hcb(D) if z = −, and W is the union of the complete (V, E)-bipartite and the complete (E, V )-bipartite digraphs if z = 1.

Spectra of digraph transformations Definitions and notations

D D00+ D-++ D10+ D+0+ D+++ D00+ D-++ D10+ D+0+ D+++

Figure: Comparing some xyz-transformations Dxyz with Dxyz

• for the

dipath D.

Spectra of digraph transformations Definitions and notations

More notations

Let D = (V, E) be a digraph in D◦ with vertex set V = {v1, . . . , vn}. A(D) = (aij): the adjacency matrix of D with aij = 1 if (vi, vj) ∈ E, and aij = 0 otherwise. A(α, D) = det(αI − A(D)): the adjacency polynomial of D. The adjacency spectrum of D is the set of roots of A(α, D) with their multiplicities. dout(v, D) = |{e ∈ E : t(e) = v}|: the out-degree of v ∈ V . din(v, D) = |{e ∈ E : h(e) = v}|: the in-degree of v ∈ V .

Spectra of digraph transformations Definitions and notations

A digraph D is balanced if din(v, D) = dout(v, D) = 0 for every v ∈ V . A digraph D is r-regular if din(v, D) = dout(v, D) = r for every v ∈ V . R(D) = diag(d1, . . . , dn): the out degree matrix of D with di = dout(vi, D). L(D) = R(D) − A(D): the Laplacian matrix of D. L(λ, D) = det(λI − L(D)): the Laplacian polynomial of D. The Laplacian spectrum of D is the set of roots of L(λ, D) with their multiplicities. A spanning ditree of D is a sub-digraph of D with a vertex r ∈ V

Spectra of digraph transformations Spectra of Dxyz for simple regular digraph D

Spectra of Dxyz for simple regular digraph D

In 1987 Zhang, Lin and Meng gave the formulas for the adjacency polynomials of D+++, D00+, D+0+, and D0++ for a general digraph D. In 2010 Liu and Meng gave the formulas for the adjacency polynomials of other Dxyz with x, y ∈ {+, −} for a simple regular digraph D.

Spectra of digraph transformations Spectra of Dxyz for simple regular digraph D

In 2013 we gave the formulas for the Laplacian polynomials (resp., the adjacency polynomials) and the number of rooted spanning ditrees of the xyz-transformations Dxyz for a simple r-regular digraph D with n vertices and all x, y, z ∈ {0, 1, +, −} in terms of r, n, and the Laplacian spectrum (resp., the adjacency spectrum)

• f D.

Spectra of digraph transformations Spectra of Dxyz

• for regular digraph D with possible loops

Spectra of Dxyz

• for regular digraph D with possible loops

We showed that the formulas for L(λ, Dxyz) with x, y, z ∈ {0, 1, +, −}, given for a regular digraph D in D, are also valid for L(λ, Dxyz

• ) where D is a regular digraph in D◦.

Spectra of digraph transformations Spectra of D[xyz] for quasi-regular digraph D

Spectra of D[xyz] for quasi-regular digraph D

A digraph D is called r-quasi-regular if D is balanced, din(v, D) ∈ {r − 1, r} for every v ∈ V (D), and D has a vertex of in-degree r − 1. If D ∈ D◦, then let ⌊D⌋ denote the digraph obtained from D by removing all loops, and let ⌈D⌉ denote the digraph obtained from D by adding a loop to every vertex without loops. Definition For an r-quasi-regular digraph D ∈ D◦, put D[xyz] = ⌊⌈D⌉xyz

• ⌋,

where x, y, z ∈ {0, 1, +, −}. The digraph D[xyz] is called the [xyz]-transformation of D.

Spectra of digraph transformations Spectra of D[xyz] for quasi-regular digraph D

We gave a procedure providing for every simple r-quasi-regular digraph D the formula for the Laplacian polynomial of D[xyz], where x, y, z ∈ {0, 1, +, −}, in terms of r, |V (D)|, and the Laplacian spectrum of D and using the formula for L(λ, Dxyz) we have given before.

Spectra of digraph transformations Spectra of D[xyz] for quasi-regular digraph D

Example

ο

a a c 2 1 c a a c 2 1 c 1 e e 3 b 2 3 3

Figure: The balanced 2-quasi-regular digraph Λ, and digraphs ⌈Λ⌉, ⌈Λ⌉−

• .

Spectra of digraph transformations Spectra of D[xyz] for quasi-regular digraph D e 2 1 e 3 a a c c s t ο b

Figure: The digraph ⌈Λ⌉−++

• .

L(λ, Λ[−++]) = L(λ, ⌈Λ⌉−++

• ) = λ(λ − 3)6(λ2 − 6λ + 7),

t(Λ[−++]) = t(⌈Λ⌉−++

• ) = 567.

Spectra of digraph transformations Spectra of F xyz for digraph-function F

Spectra of F xyz for digraph-function F

A digraph D = (V, E) is called a digraph-function if there exists a function f : V → V such that (x, y) ∈ E if and only if y = f(x). The inverse D−1 of a digraph D = (V, E): a digraph with V (D−1) = V and E(D−1) = {(y, x) : (x, y) ∈ E}. CF is the set of connected digraph-functions and their inverses. Obviously, if F ∈ CF, then F has a unique cycle C. Let c(F) denote the number of vertices of C.

Spectra of digraph transformations Spectra of F xyz for digraph-function F

Let F be a connected digraph-function or its inverse. We gave explicit formula for every A(α, F xyz) in terms of the adjacency spectrum of F, or equivalently, in terms of |V (F)| and c(F) if z = 0 or z ∈ {1, +} and x, y ∈ {0, z}. We also gave explicit formula for every L(λ, F xyz) in terms of the Laplacian spectrum of F, or equivalently, in terms of |V (F)| and c(F) if z = 0, 1, or z = + and x, y ∈ {0, +}, or z = − and x, y ∈ {1, −}.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Isomorphic and cospectral non-isomorphic xyz-transformations

Theorem (Deng and Kelmans, 2013) Let D be a digraph with possible parallel arcs or loops. Then

• D+0z and D0+z are adjacency cospectral digraphs for z ∈ {0, +}

and if D ∼ = Dℓ then D+0z ∼ = D0+z.

• If D is r-regular, then D+0z and D0+z are adjacency cospectral

digraphs for z ∈ {1, −} and D+0z ∼ = D0+z for r > 1.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Theorem (Deng and Kelmans 2013) Let F be a digraph-function or its inverse. Then

1 F xyz and F yxz are isomorphic for all x, y ∈ {0, 1, +, −} and

z ∈ {0, 1},

2 if F is 1-regular, then F xyz and F yxz are isomorphic for all

x, y, z ∈ {0, 1, +, −}, and

3 F xyz and F yxz are adjacency cospectral and non-isomorphic

for all x, y, z ∈ {0, 1, +, −} if and only if z ∈ {+, −}, x = y, and F is not regular.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Theorem (Deng and Kelmans) Let F be a digraph-function and x, y, z ∈ {0, 1, +, −}. Then F xyz and F yxz are non-isomorphic and Laplacian cospectral if and only if F is non-regular, x = y and z ∈ {+, −}. Corollary Let F be a digraph-function, x, y, z ∈ {0, 1, +, −}, and F xyz and F yxz are not isomorphic. Then F xyz and F yxz are Laplacian cospectral if and only if they are adjacency cospectral.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Cospectral xyz-transformations for digraphs having possible loops

Theorem (Deng and Kelmans) Let D be a simple digraph. Then

1 L(λ, Dxyz

• ) = L(λ, Dxyz) (i.e., Dxyz
• and Dxyz are Laplacian

cospectral ) for all x, y, z ∈ {0, 1, +, −}, and

2 Dxyz and Dxyz

• are not isomorphic if and only if

{−, 1} ∩ {x, y} = ∅.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

More constructions on cospectral non-isomorphic and non-regular digraphs.

Let D and D′ be disjoint digraphs, X ⊆ D, X′ ⊆ D′, X = ∅, and π a bijection from X to X′. Let DXπX′D′ denote the digraph

• btained from D and D′ by identifying vertex x in D with the

vertex π(x) in D′ for every x ∈ X. Given a digraph D, let Vin(D) = {v ∈ V (D) : dout(v) = 0} and Vout(D) = {v ∈ V (D) : din(v) = 0}. A digraph D is called acyclic if D has no directed cycles.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Using the above notation let F = DXπX′D′, where D and D′ are disjoint, D′ is acyclic, and X′ ⊆ Vin(D′) or X′ ⊆ Vout(D′). Theorem (Deng and Kelmans) Denote n = v(F), k = v(D). Then A(λ, F) = λn−kA(λ, D). Theorem (Deng and Kelmans) Suppose that |V (D′) \ X′| = e(D) − v(D). Then A(λ, Dl) = A(λ, F). Theorems above give constructions that provide an infinite variety

• f non-isomorphic adjacency cospectral digraphs.

Spectra of digraph transformations Isomorphic and cospectral non-isomorphic xyz-transformations

Theorem (Deng and Kelmans) Let D and D′ be disjoint digraphs, F = DXπX′D′, and n = v(F), k = v(D). Suppose that D′ is an acyclic digraph and X′ ⊆ Vin(D′). Then L(λ, F) = (λ − 1)n−kL(λ, D). This theorem gives a construction that provides an infinite variety

• f pairs of non-isomorphic Laplacian cospectral digraphs.

Spectra of digraph transformations References

References I

• A. Deng, A. Kelmans, and J. Meng, Laplacian spectra of regular graph

transformations, Discrete Applied Mathematics 161 (2013) 118-133.

• A. Deng and A. Kelmans, Spectra of digraph transformations, Linear

Algebra Appl. 439 (2013) 106-132.

• A. Deng and A. Kelmans, Laplacian spectra of digraph transformations,

manscript.

• J. Liu and J. Meng, Spectra of transformation digraphs of regular digraph,

Linear Multilinear Algebra 58 (2010) 555-561.

• J. Yan, K. Xu, Spectra of transformation graphs of regular graph, Applied

Mathematics, A Journal of Chinese Universities (Ser.A) 23 (2008) 476-480.

Spectra of digraph transformations References

References II

• F. Zhang, G. Lin, and J. Meng, The characteristic polynomials of digraphs

formed by some unary operations, J. Xinjiang Univ. (Natural Science Edition) 4 (1987) 1-6.

Spectra of digraph transformations References

Thank you !