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### Remarks on distance-regular graphs

Mitsuo Yoshizawa
1981 Discrete Mathematics
Let r be a 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.  ...

### A remark on the intersection arrays of distance-regular graphs

Alan Boshier, Kazumasa Nomura
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 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).  ...

### A short note on a short remark of Graham and Lovász [article]

Jernej Azarija
2013 arXiv   pre-print
A large representative of this class being the Paley graphs.The result is obtained by derving the eigenvalues of the distance matrix of a strongly-regular graph.  ...  It was remarked in  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.  ...

### On the new extension of distance-balanced graphs

Morteza Faghani, Ehsan Pourhadi, Hassan kharazi
2016 Transactions on Combinatorics
In our consideration‎, ‎we also define the new concept locally 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

Leonard H. Soicher
1995 Electronic Journal of Combinatorics
We describe a new 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  , Brouwer, Cohen and Neumaier discuss many distance-regular graphs related to the famous Golay codes. In this note, we describe yet another such graph.  ...

### On Expanders Graphs: Parameters and Applications [article]

H. L. Janwa A. K. Lal
2004 arXiv   pre-print
We give a new lower bound 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.  ...

### On some distance-regular graphs with many vertices [article]

Dean Crnkovic, Sanja Rukavina, Andrea Svob
2018 arXiv   pre-print
We construct 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  for more information.  ...

### Spectra of strongly Deza graphs [article]

2021 arXiv   pre-print
In this paper we give a spectral characterisation of strongly Deza 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  and the survey  .  ...

### Geometric distance-regular graphs without 4-claws [article]

Sejeong Bang
2011 arXiv   pre-print
In this paper, we determine the non-complete 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.  ...

### A new distance-regular graph of diameter 3 on 1024 vertices [article]

Minjia Shi, Denis Krotov, Patrick Solé
2018 arXiv   pre-print
Its coset 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.  ...

### Distance-regular graphs obtained from the Mathieu groups [article]

Dean Crnkovic, Nina Mostarac, Andrea Svob
2021 arXiv   pre-print
In this paper we construct 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.  ...

### Linear programming bounds for regular graphs [article]

Hiroshi Nozaki
2015 arXiv   pre-print
The known 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.  ...

### On bipartite distance-regular Cayley graphs with diameter 3 [article]

Mojtaba Jazaeri
2021 arXiv   pre-print
In this paper, we show that every bipartite 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.  ...

### Linear Programming Bounds for Regular Graphs

Hiroshi Nozaki
2015 Graphs and Combinatorics
The known 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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