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Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2019.2.32115 On a DistanceRegular Graph with an Intersection Array {35,28,6;1,2,30}
Abstract:
It is proved that for a distanceregular graph \(\Gamma\) of diameter 3 with eigenvalue \(\theta_2=1\) the complement graph of \(\Gamma_3\) is pseudogeometric for \(pG_{c_3}(k,b_1/c_2 )\). Bang and Koolen investigated distanceregular graphs with intersection arrays \({(t+1)s,ts, (s+1\psi); 1,2,(t+1)\psi}\). If \(t=4\), \(s=7\), \(\psi=6\) then we have array \({35,28,6;1,2,30}\). Distanceregular graph \(\Gamma\) with intersection array \(\{35,28,6; 1,2,30\}\) has spectrum of \(35^1\), \(9^{168}\), \(1^{182}\), \(5^{273}\), \(v=1+35+490+98=624\) vertices and \(\overline{\Gamma}_3\) is a pseudogeometric graph for \(pG_{30}(35,14)\). Due to the border of Delsarte, the order of clicks in \(\Gamma\) is not more than 8. It is also proved that either a neighborhood of any vertex in \(\Gamma\) is the union of an isolated 7click, or the neighborhood of any vertex in \(\Gamma\) does not contain a 7click and is a connected graph. The structure of the group \(G\) of automorphisms of a graph \(\Gamma\) with an intersection array \(\{35,28,6; 1,2,30\}\) has been studied. In particular, \(\pi(G)\subseteq\{2,3,5,7,13\}\) and the edge symmetric graph \(\Gamma\) has a solvable group automorphisms.
Keywords: distanceregular graph, Delsarte clique, geometric graph
Language: Russian
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For citation: Makhnev, A. A. and Tokbaeva, A. A. On a DistanceRegular Graph with an Intersection Array {35,28,6;1,2,30}, Vladikavkaz Math. J., 2019, vol. 21, no. 2, pp. 2737 (in Russian). DOI 10.23671/VNC.2019.2.32115 ← Contents of issue 
 

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