Geodesic
To the theory of Shilla graphs with $b_2=c_2$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1135-1146.

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In this paper by using exact formulas for multiplicities of eigenvalues it is founded new infinite serie intersection arrays of $Q$-polynomial Shilla graph with $b_2 = c_2$. Intersection array of $Q$-polynomial Shilla graph $\Gamma$ with $b_2=c_2$ is $\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}$ and for any vertex $u\in \Gamma$ the subgraph $\Gamma_3(u)$ is an antipodal distance-regular graph with the intersection array $\{t(2r+1),(2r-1)(t+1),1;1,t+1,t(2r+1)\}$. In case $t=2r^2-1$ the intersection array is feasible and in case $t=r(2lr-(l+1))$ the intersection array is feasible only if $(l,r)\in \{(1,2),(2,1),(4,1),(6,1)\}$.
Keywords: distance-regular graph, Shilla graph. 
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     title = {To the theory of {Shilla} graphs with $b_2=c_2$},
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A. A. Makhnev; I. N. Belousov. To the theory of Shilla graphs with $b_2=c_2$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1135-1146. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a78/

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