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Remarks on distance-regular graphs

1981
*
Discrete Mathematics
*

Let r be a

doi:10.1016/0012-365x(81)90027-3
fatcat:74k5fttlijhd5gjlizmnfpg6ge
*distance*-*regular*gvayh stihsfying eit! ... Let r be a*distance*-*regular**graph*[ 1] with valency li (23), diameter d, girth g. For u~Vr ( and iE(O,l,...,d}, we define ri(U)=(UEVrla(u,U)=i}. Theorem. ...##
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A remark on the intersection arrays of distance-regular graphs

1988
*
Journal of combinatorial theory. Series B (Print)
*

It is shown that the number of columns of type (1, 1, k-2) in the intersection array of a

doi:10.1016/0095-8956(88)90084-6
fatcat:uadaalhwhvcelmlmv724oqnh7u
*distance*-*regular**graph*with valency k and girth >3 is at most four. ... PROOF OF THEOREM Let G = (V, E) be a*distance*-*regular**graph*with valency k 2 3. ... If pJ(u, u) depends only*on*the*distance*between u and v, rather than the individual vertices, then G is said to be*distance*-*regular*. In this case, we write pj, = pi(u, v), where m = a(u, u). ...##
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A short note on a short remark of Graham and Lovász
[article]

2013
*
arXiv
*
pre-print

A large representative of this class being the Paley graphs.The result is obtained by derving the eigenvalues of the

arXiv:1303.4517v2
fatcat:6youdvf5vnbhvkzzu4b447yg3y
*distance*matrix of a strongly-*regular**graph*. ... It was*remarked*in [1] that it is not known whether there is a*graph*for which np(G) > nn (G). ... As*remarked*at the begining of the section a computer search indicated that there is no optimistic*graph**on*at most 11 vertices. ...##
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On the new extension of distance-balanced graphs

2016
*
Transactions on Combinatorics
*

In our consideration, we also define the new concept locally

doaj:d3070ddd75644b579aac80da3d0795e1
fatcat:jo3fuabhxvgpzg5uejgj3vexx4
*regularity*in order to find a connection between n n-*distance*-balanced*graphs*and their lexicographic product. ... Moreover, we conclude a connection between*distance*-balanced and 2-*distance*-balanced*graphs*. ... Acknowledgement We are grateful to the referees for their suggestions and helpful*remarks*. ...##
###
Yet Another Distance Regular Graph Related to a Golay Code

1995
*
Electronic Journal of Combinatorics
*

We describe a new

doi:10.37236/1220
fatcat:zvxu5rmf3jcbtmjty7lga3oa7a
*distance*-*regular*, but not*distance*-transitive,*graph*. This*graph*has intersection array $\{110,81,12;1,18,90\}$, and automorphism group $M_{22}\colon 2$. ... scheme to determine if it came from a*distance*-*regular**graph*. ... In [1] , Brouwer, Cohen and Neumaier discuss many*distance*-*regular**graphs*related to the famous Golay codes. In this note, we describe yet another such*graph*. ...##
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On Expanders Graphs: Parameters and Applications
[article]

2004
*
arXiv
*
pre-print

We give a new lower bound

arXiv:cs/0406048v1
fatcat:n2x7eoqerfbdfkkjbwbktzadrq
*on*the expansion coefficient of an edge-vertex*graph*of a d-*regular**graph*. ... As a consequence, we obtain an improvement*on*the lower bound*on*relative minimum*distance*of the expander codes constructed by Sipser and Spielman. ... δ, i.e., Proposition 2. 1 . 1 Let G be a d−*regular**graph**on*n vertices. ...##
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On some distance-regular graphs with many vertices
[article]

2018
*
arXiv
*
pre-print

We construct

arXiv:1809.10197v2
fatcat:r3y2ikbxg5cjjamlqby76m4d6m
*distance*-*regular**graphs*, including strongly*regular**graphs*, admitting a transitive action of the Chevalley groups G_2(4) and G_2(5), the orthogonal group O(7,3) and the Tits group T=^2F_4( ... Most of the constructed*graphs*have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained*graphs*are new. ...*Remark*5 The*distance*-*regular**graphs*Γ 2 7 and Γ 2 8 have diameter 3. They belong to the family of*graphs*of Lie type from Chevalley groups. See [5] for more information. ...##
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Spectra of strongly Deza graphs
[article]

2021
*
arXiv
*
pre-print

In this paper we give a spectral characterisation of strongly Deza

arXiv:2101.06877v1
fatcat:ehvutd7tuvdxvlfnryjvz35n44
*graphs*, show relationships between eigenvalues, and study strongly Deza*graphs*which are*distance*-*regular*. ... A strongly Deza*graph*is a Deza*graph*with strongly*regular*children. ...*Distance*-*regular*Deza*graphs*Now we investigate when a*distance*-*regular**graph*can be a strongly Deza*graph*. For background*on**distance*-*regular**graphs*, we refer to the book [2] and the survey [7] . ...##
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Geometric distance-regular graphs without 4-claws
[article]

2011
*
arXiv
*
pre-print

In this paper, we determine the non-complete

arXiv:1101.0440v1
fatcat:twgdcc6cwzckhiciq4wmswgcpu
*distance*-*regular**graphs*satisfying {3, 8/3(a_1+1)}<k<4a_1+10-6c_2. ... To prove this result, we first show by considering non-existence of 4-claws that any non-complete*distance*-*regular**graph*satisfying {3, /3(a_1+1)}<k<4a_1+10-6c_2 is a geometric with smallest eigenvalue ... From now*on*, we assume a 1 ≥ 2. First suppose c h+1 ≥ 2, where h = h(Γ) is the head of Γ in (2) . Then by (9) We*remark**on*the*distance*-*regular**graphs*in Theorem 4.3. ...##
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A new distance-regular graph of diameter 3 on 1024 vertices
[article]

2018
*
arXiv
*
pre-print

Its coset

arXiv:1806.07069v2
fatcat:eyrrl7wfpbb4pl3zipeisfp274
*graph*is*distance*-*regular*of diameter three*on*2^10 vertices, with new intersection array {33,30,15;1,2,15}. The automorphism groups of the code, and of the*graph*, are determined. ... Connecting the vertices at*distance*two gives a strongly*regular**graph*of (previously known) parameters (2^10,495,238,240). ... The case m = 2 of the above parameters leads to a putative coset*graph*listed in [6, p.428 ] as a*distance*-*regular**graph**on*1024 = 4 5 vertices of diameter 3. ...##
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Distance-regular graphs obtained from the Mathieu groups
[article]

2021
*
arXiv
*
pre-print

In this paper we construct

arXiv:2101.02790v2
fatcat:osn36ejbdrac5prykhv3pmqk2i
*distance*-*regular**graphs*admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups M_11, M_12, M_22, M_23 and M_24. ... From the code spanned by the adjacency matrix of the strongly*regular**graph*with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite ... The authors would like to thank Sven Reichard for pointing out that the*graphs*Γ 1 3 and Γ 2 5 constructed in this paper are not arising from orthogonal arrays. ...##
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Linear programming bounds for regular graphs
[article]

2015
*
arXiv
*
pre-print

The known

arXiv:1407.4562v2
fatcat:l2forhw225ab5b2kzae7f2bmzy
*graphs*satisfying g>2d-1 are Moore*graphs*, incidence*graphs*of*regular*generalized polygons of order (s,s), triangle-free strongly*regular**graphs*, and the odd*graph*of degree 4. ... As an application of this bound, we prove that a connected k-*regular**graph*satisfying g>2d-1 has the minimum second-largest eigenvalue of all k-*regular**graphs*of the same size, where d is the number of ... For connected*regular**graph*, we have a very similar situation to the above argument*on*the sphere. Let G be a connected k-*regular**graph*with v vertices. ...##
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On bipartite distance-regular Cayley graphs with diameter 3
[article]

2021
*
arXiv
*
pre-print

In this paper, we show that every bipartite

arXiv:1904.06696v3
fatcat:oiypidvglzeidemmfns7mn35qq
*distance*-*regular*Cayley*graph*with diameter 3 can be constructed*on*the semidirect product of a group and ℤ_2, except possibly for*one*case. ... The author would like to thank professorŠtefko Miklavič for his valuable comments*on*the first draft of this paper and also professor Edwin R. van Dam for his valuable comments which improved the presentation ...*Remark*3.2. Let Γ be a bipartite*distance*-*regular**graph*with diameter 3 and valency k. ...##
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Linear Programming Bounds for Regular Graphs

2015
*
Graphs and Combinatorics
*

The known

doi:10.1007/s00373-015-1613-7
fatcat:yi2xprr6pvhrvli7ycxo5eouba
*graphs*satisfying g > 2d − 1 are Moore*graphs*, incidence*graphs*of*regular*generalized polygons of order (s, s), triangle-free strongly*regular**graphs*, and the odd*graph*of degree 4. ... As an application of this bound, we prove that a connected k-*regular**graph*satisfying g > 2d − 1 has the minimum second-largest eigenvalue of all k-*regular**graphs*of the same size, where d is the number ... For connected*regular**graph*, we have a very similar situation to the above argument*on*the sphere. Let G be a connected k-*regular**graph*with v vertices. ...##
###
Page 7198 of Mathematical Reviews Vol. , Issue 95m
[page]

1995
*
Mathematical Reviews
*

Andrew Woldar (1-VLNV; Villanova, PA)
95m:05253 05E30
Wajima, Masayuki
A

*remark**on**distance*-*regular**graphs*with a circuit of diameter ¢ + |. (English summary) Math. Japon. 40 (1994), no. 3, 433-437. ... The author determines conditions under which the merging of the first and last classes of an antipodal*distance*-*regular**graph*leads again to a*distance*-*regular**graph*. ...
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