Distance mean-regular graphs
We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be a graph with vertex set V , diameter D , adjacency matrix A , and adjacency algebra A . Then, Γ is distance mean-regular when, for a given u...
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| Published in: | Designs, codes, and cryptography Vol. 84; no. 1-2; pp. 55 - 71 |
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| Language: | English |
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01.07.2017
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| Abstract | We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let
Γ
be a graph with vertex set
V
, diameter
D
, adjacency matrix
A
, and adjacency algebra
A
. Then,
Γ
is
distance mean-regular
when, for a given
u
∈
V
, the averages of the intersection numbers
p
i
j
h
(
u
,
v
)
=
|
Γ
i
(
u
)
∩
Γ
j
(
v
)
|
(number of vertices at distance
i
from
u
and distance
j
from
v
) computed over all vertices
v
at a given distance
h
∈
{
0
,
1
,
…
,
D
}
from
u
, do not depend on
u
. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of
Γ
and, hence, they generate a subalgebra of
A
. Some other algebras associated to distance mean-regular graphs are also characterized. |
|---|---|
| AbstractList | We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let
Γ
be a graph with vertex set
V
, diameter
D
, adjacency matrix
A
, and adjacency algebra
A
. Then,
Γ
is
distance mean-regular
when, for a given
u
∈
V
, the averages of the intersection numbers
p
i
j
h
(
u
,
v
)
=
|
Γ
i
(
u
)
∩
Γ
j
(
v
)
|
(number of vertices at distance
i
from
u
and distance
j
from
v
) computed over all vertices
v
at a given distance
h
∈
{
0
,
1
,
…
,
D
}
from
u
, do not depend on
u
. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of
Γ
and, hence, they generate a subalgebra of
A
. Some other algebras associated to distance mean-regular graphs are also characterized. We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be a graph with vertex set V, diameter D, adjacency matrix A , and adjacency algebra A . Then, Γ is distance mean-regular when, for a given u ∈ V , the averages of the intersection numbers p i j h ( u , v ) = | Γ i ( u ) ∩ Γ j ( v ) | (number of vertices at distance i from u and distance j from v) computed over all vertices v at a given distance h ∈ { 0 , 1 , … , D } from u, do not depend on u. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of Γ and, hence, they generate a subalgebra of A . Some other algebras associated to distance mean-regular graphs are also characterized. We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let G be a graph with vertex set V , diameter D, adjacency matrix A, and adjacency algebra A. Then, G is distance mean-regular when, for a given u ¿ V , the averages of the intersection numbers p h ij (u, v) = |Gi(u) n Gj (v)| (number of vertices at distance i from u and distance j from v) computed over all vertices v at a given distance h ¿ {0, 1, . . . , D} from u, do not depend on u. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of G and, hence, they generate a subalgebra of A. Some other algebras associated to distance mean-regular graphs are also characterized. Peer Reviewed |
| Author | Diego, V. Fiol, M. A. |
| Author_xml | – sequence: 1 givenname: V. surname: Diego fullname: Diego, V. organization: Departamento de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya – sequence: 2 givenname: M. A. surname: Fiol fullname: Fiol, M. A. email: fiol@ma4.upc.edu organization: Departamento de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Barcelona Graduate School of Mathematics |
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| Keywords | 05E30 Vertex-transitive graph 05C50 Interlacing Adjacency Algebra Distance-regular graph Intersection mean-matrix Distance mean-regular graph Spectrum |
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| References | Godsil C.D., Shawe-Taylor J.: Distance-regularised graphs are distance-regular or distance-biregular. J. Comb. Theory Ser. B 43(1), 14–24 (1987). Hilano T., Nomura K.: Distance degree regular graphs. J. Comb. Theory Ser. B 37, 96–100 (1984). Fiol M.A.: Pseudo-distance-regularized graphs are distance-regular or distance-biregular. Linear Algebra Appl. 437, 2973–2977 (2012). Biggs N.: Intersection matrices for linear graphs. In: Welsh D.J.A. (ed.) Combinatorial Mathematics and Its Applications, pp. 15–23. Academic Press, London (1971). Weichsel P.: On distance-regularity in graphs. J. Comb. Theory Ser. B 32, 156–161 (1982). Haemers W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226–228, 593–616 (1995). Hazewinkel M., Gubareni N., Gubareni N.M., Kirichenko V.V.: Algebras, Rings and Modules, vol. 1. Springer, Heildelberg (2004). Fiol M.A.: Eigenvalue interlacing and weight parameters of graphs. Linear Algebra Appl. 290, 275–301 (1999). Biggs N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974), 2nd edn. (1993). Abiad A., van Dam E.R., Fiol M.A.: Some spectral and quasi-spectral characterizations of distance-regular graphs. Preprint, arXiv:1404.3973v3 [math.CO], (2014). Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993). Bannai E., Ito T.: Algebraic Combinatorics I: Association Schemes. Benjamin Cummings Lecture Notes Series, vol. 58. Benjammin-Cummings, London (1993). Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York. http://homepages.cwi.nl/~aeb/math/ipm/ (2012). van Dam E.R., Koolen J.H., Tanaka H.: Distance-regular graphs. Preprint arXiv:1410.6294 [math.CO]. (2014). Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989). Fiol M.A., Garriga E.: From local adjacency polynomials to locally pseudo-distance-regular graphs. J. Comb. Theory Ser. B 71, 162–183 (1997). Cámara M., Fàbrega J., Fiol M.A., Garriga E.: Some families of orthogonal polynomials of a discrete variable and their applications to graphs ans codes. Electron. J. Comb. 16, R83 (2009). 208_CR5 208_CR4 208_CR7 208_CR6 208_CR9 208_CR8 208_CR17 208_CR16 208_CR15 208_CR14 208_CR13 208_CR12 208_CR11 208_CR1 208_CR10 208_CR3 208_CR2 |
| References_xml | – reference: Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs. Springer, Berlin (1989). – reference: van Dam E.R., Koolen J.H., Tanaka H.: Distance-regular graphs. Preprint arXiv:1410.6294 [math.CO]. (2014). – reference: Haemers W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226–228, 593–616 (1995). – reference: Hilano T., Nomura K.: Distance degree regular graphs. J. Comb. Theory Ser. B 37, 96–100 (1984). – reference: Cámara M., Fàbrega J., Fiol M.A., Garriga E.: Some families of orthogonal polynomials of a discrete variable and their applications to graphs ans codes. Electron. J. Comb. 16, R83 (2009). – reference: Fiol M.A.: Eigenvalue interlacing and weight parameters of graphs. Linear Algebra Appl. 290, 275–301 (1999). – reference: Fiol M.A.: Pseudo-distance-regularized graphs are distance-regular or distance-biregular. Linear Algebra Appl. 437, 2973–2977 (2012). – reference: Hazewinkel M., Gubareni N., Gubareni N.M., Kirichenko V.V.: Algebras, Rings and Modules, vol. 1. Springer, Heildelberg (2004). – reference: Biggs N.: Intersection matrices for linear graphs. In: Welsh D.J.A. (ed.) Combinatorial Mathematics and Its Applications, pp. 15–23. Academic Press, London (1971). – reference: Weichsel P.: On distance-regularity in graphs. J. Comb. Theory Ser. B 32, 156–161 (1982). – reference: Bannai E., Ito T.: Algebraic Combinatorics I: Association Schemes. Benjamin Cummings Lecture Notes Series, vol. 58. Benjammin-Cummings, London (1993). – reference: Fiol M.A., Garriga E.: From local adjacency polynomials to locally pseudo-distance-regular graphs. J. Comb. Theory Ser. B 71, 162–183 (1997). – reference: Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993). – reference: Godsil C.D., Shawe-Taylor J.: Distance-regularised graphs are distance-regular or distance-biregular. J. Comb. Theory Ser. B 43(1), 14–24 (1987). – reference: Brouwer A.E., Haemers W.H.: Spectra of Graphs. Springer, New York. http://homepages.cwi.nl/~aeb/math/ipm/ (2012). – reference: Abiad A., van Dam E.R., Fiol M.A.: Some spectral and quasi-spectral characterizations of distance-regular graphs. Preprint, arXiv:1404.3973v3 [math.CO], (2014). – reference: Biggs N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974), 2nd edn. (1993). – ident: 208_CR5 – ident: 208_CR11 – ident: 208_CR2 – ident: 208_CR6 – ident: 208_CR3 doi: 10.1007/978-3-642-74341-2 – ident: 208_CR7 – ident: 208_CR16 – ident: 208_CR14 – ident: 208_CR1 – ident: 208_CR8 doi: 10.1006/jctb.1997.1778 – ident: 208_CR12 doi: 10.1016/0095-8956(87)90027-X – ident: 208_CR9 doi: 10.1016/S0024-3795(98)10238-0 – ident: 208_CR10 doi: 10.1016/j.laa.2012.07.019 – ident: 208_CR4 doi: 10.1007/978-1-4614-1939-6 – ident: 208_CR17 doi: 10.1016/0095-8956(82)90031-4 – ident: 208_CR13 doi: 10.1016/0024-3795(95)00199-2 – ident: 208_CR15 doi: 10.1016/0095-8956(84)90050-9 |
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| Snippet | We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let
Γ
be... We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let Γ be... We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let G be... |
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| SubjectTerms | 05 Combinatorics 05C Graph theory 05E Algebraic combinatorics Adjacency Algebra Algebra Circuits Classificació AMS Coding and Information Theory Combinacions (Matemàtica) Combinatorial analysis Combinatòria Computation Computer Science Cryptology Data Structures and Information Theory Discrete Mathematics in Computer Science Distance mean-regular graph Distance-regular graph Grafs, Teoria de Graph theory Graphs Information and Communication Interlacing Intersection mean-matrix Matemàtica discreta Matemàtiques i estadística Spectrum Teoria de grafs Vertex-transitive graph Àrees temàtiques de la UPC |
| Title | Distance mean-regular graphs |
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