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 on 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 D xyz for simple regular digraph D

Spectra of digraph transformations Contents 1 Definitions and notations 2 Spectra of D xyz for simple regular digraph D 3 Spectra of D xyz for regular digraph D with possible loops ◦

Spectra of digraph transformations Contents 1 Definitions and notations 2 Spectra of D xyz for simple regular digraph D 3 Spectra of D xyz 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 D xyz for simple regular digraph D 3 Spectra of D xyz 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 D xyz for simple regular digraph D 3 Spectra of D xyz 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 D xyz for simple regular digraph D 3 Spectra of D xyz 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 . D 0 is the digraph with V ( D 0 ) = V and with no arcs. D 1 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 ◦ . D 0 ◦ is the digraph with V ( D 0 ◦ ) = V and with no arcs. D 1 ◦ 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 . D ℓ is the line digraph of D , with V ( D ℓ ) = E ( D ) and E ( D ℓ ) = { ( 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 only if v ∈ V, e ∈ E and vertex v = t ( e ) (resp. v � = t ( e ) ) in D . H ( D ) (resp. H cb ( D ) ) is the digraph with vertex set V ∪ E and such that ( e, v ) is an arc in H ( D ) (resp. in H cb ( 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 ) H ( D ) T cb ( D ) H cb ( D ) Figure: Digraphs T ( D ) , H ( D ) , T cb ( D ) , and H cb ( 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 D xyz of D is the digraph such that D xy 0 = D x ∪ ( D l ) y and D xyz = D xy 0 ∪ W , where W = T ( D ) ∪ H ( D ) if z = + , W = T cb ( D ) ∪ H cb ( 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 00+ D +0+ D 10+ D +++ D -++ D D 11- D 01- D +-- D -1- 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 D xyz of D is the digraph such that ◦ ◦ ∪ ( D l ) y and D xyz D xy 0 = D xy 0 = D x ∪ W , where ◦ ◦ ◦ W = T ( D ) ∪ H ( D ) if z = + , W = T cb ( D ) ∪ H cb ( 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 00+ D +0+ D 10+ D +++ D -++ D D 00+ D +0+ D 10+ D +++ D -++ Figure: Comparing some xyz -transformations D xyz with D xyz 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 = { v 1 , . . . , v n } . A ( D ) = ( a ij ) : the adjacency matrix of D with a ij = 1 if ( v i , v j ) ∈ E , and a ij = 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. d out ( v, D ) = |{ e ∈ E : t ( e ) = v }| : the out-degree of v ∈ V . d in ( 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 d in ( v, D ) = d out ( v, D ) � = 0 for every v ∈ V . A digraph D is r -regular if d in ( v, D ) = d out ( v, D ) = r for every v ∈ V . R ( D ) = diag( d 1 , . . . , d n ) : the out degree matrix of D with d i = d out ( v i , 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 D xyz for simple regular digraph D Spectra of D xyz for simple regular digraph D In 1987 Zhang, Lin and Meng gave the formulas for the adjacency polynomials of D +++ , D 00+ , D +0+ , and D 0++ for a general digraph D . In 2010 Liu and Meng gave the formulas for the adjacency polynomials of other D xyz with x, y ∈ { + , −} for a simple regular digraph D .

Spectra of digraph transformations Spectra of D xyz 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 D xyz 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) of D .

Spectra of digraph transformations Spectra of D xyz for regular digraph D with possible loops ◦ Spectra of D xyz for regular digraph D with possible loops ◦ We showed that the formulas for L ( λ, D xyz ) with x, y, z ∈ { 0 , 1 , + , −} , given for a regular digraph D in D , are also valid for L ( λ, D xyz ) 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, d in ( 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, D xyz ) we have given before.

Spectra of digraph transformations Spectra of D [ xyz ] for quasi-regular digraph D Example 1 1 1 a a a a b e e 3 3 3 c c c c 2 2 2 ο Figure: The balanced 2-quasi-regular digraph Λ , and digraphs ⌈ Λ ⌉ , ⌈ Λ ⌉ − ◦ .