Geodesic
Distance-Regular Shilla Graphs with~$b_2=c_2$
Matematičeskie zametki, Tome 103 (2018) no. 5, pp. 730-744.

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A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigenvalue $\theta_1$ equal to $a_3$. For a Shilla graph, let us put $a=a_3$ and $b=k/a$. It is proved in this paper that a Shilla graph with $b_2=c_2$ and noninteger eigenvalues has the following intersection array: $$ \biggl\{\frac{b^2(b-1)}2\mspace{2mu}, \frac{(b-1)(b^2-b+2)}2\mspace{2mu}, \frac{b(b-1)}4\mspace{2mu};1, \frac{b(b-1)}4\mspace{2mu}, \frac{b(b-1)^2}2\biggr\}. $$ If $\Gamma$ is a $Q$-polynomial Shilla graph with $b_2=c_2$ and $b=2r$, then the graph $\Gamma$ has intersection array $$ \{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\} $$ and, for any vertex $u$ in $\Gamma$, the subgraph $\Gamma_3(u)$ is an antipodal distance-regular graph with intersection array $$ \{t(2r+1),(2r-1)(t+1),1;1,t+1,t(2r+1)\}. $$ The Shilla graphs with $b_2=c_2$ and $b=4$ are also classified in the paper.
Keywords: distance-regular graph, Shilla graph
Mots-clés : graph automorphism. 
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     author = {A. A. Makhnev and M. S. Nirova},
     title = {Distance-Regular {Shilla} {Graphs} with~$b_2=c_2$},
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A. A. Makhnev; M. S. Nirova. Distance-Regular Shilla Graphs with~$b_2=c_2$. Matematičeskie zametki, Tome 103 (2018) no. 5, pp. 730-744. http://geodesic.mathdoc.fr/item/MZM_2018_103_5_a7/

[1] J. H. Koolen, J. Park, “Shilla distance-regular graphs”, European J. Combin., 31:8 (2010), 2064–2073 | DOI | MR | Zbl

[2] A. A. Makhnev, D. V. Paduchikh, “O gruppe avtomorfizmov distantsionno regulyarnogo grafa s massivom peresechenii $\{24,21,3;1,3,18\}$”, Algebra i logika, 51:4 (2012), 476–495 | MR | Zbl

[3] A. Jurišić, J. Vidali, “Extremal 1-codes in distance-regular graphs of diameter 3”, Des. Codes Cryptogr., 65:1-2 (2012), 29–47 | DOI | MR | Zbl

[4] J. Vidali, Kode v razdaljno regularnih grafih, Doctorska Dissertacija, Univerza v Ljubljani, Ljubljana, 2013

[5] A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989 | MR | Zbl

[6] J. H. Koolen, J. Park, H. Yu, “An inequality involving the second largest and smallest eigenvalue of a distance-regular graph”, Linear Algebra Appl., 434:12 (2011), 2404–2412 | DOI | MR | Zbl

[7] J. H. Koolen, J. Park, “Distance-regular graphs with large $a_1$ or $c_2$ at least half the valency”, J. Combin. Theory Ser. A, 119:3 (2012), 546–555 | DOI | MR | Zbl