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Antipodal Krein Graphs and Distance-Regular Graphs Close to Them
Doklady Mathematics ( IF 0.5 ) Pub Date : 2020-09-10 , DOI: 10.1134/s1064562420030138
A. A. Makhnev

Abstract

An antipodal nonbipartite distance-regular graph Γ of diameter 3 has an intersection array \(\{ k,(r - 1){{c}_{2}},\;1;\;1,\;{{c}_{2}},\;k\} \) (\({{c}_{2}} < k - 1\)) and eigenvalues k, n, –1, and –m, where n and –m are the roots of the quadratic equation \({{x}^{2}} - ({{a}_{1}} - {{c}_{2}})x - k = 0\). The Krein bound \(q_{{33}}^{3} \geqslant 0\) gives \(m \leqslant {{n}^{2}}\) if \(r \ne 2\). In the case \(m = {{n}^{2}}\), following Godsil, we call Γ an antipodal Krein graph. The point graph Σ of \(GQ(q,\;{{q}^{2}})\) having spread gives an antipodal Krein graph with \(r = q + 1\). If Σ has an automorphism σ of order f that fixes every component of the spread, then the graph \(\bar {\Sigma } = \Sigma {\text{/}}\langle \sigma \rangle \) whose vertices are σ-orbits on a point set and two orbits are adjacent if a vertex of one orbit is adjacent to a vertex of the other is a distance-regular graph with intersection array {q3, \(((q + 1){\text{/}}f - 1)({{q}^{2}} - 1)f,\;1;\;1,\;({{q}^{2}} - 1)f,\;{{q}^{3}}\} \) and every local subgraph Δ(u) is pseudogeometric for \(p{{G}_{{f - 1}}}(q - 1,\;(q + 1)(f - 1))\). If f = 2, then we have a pseudogeometric graph for \(GQ(q - 1,\;q + 1)\). Hence, a locally pseudo GQ(4, 6) graph with intersection array {125, 96, 1; 1, 48, 125} and a locally pseudo GQ(6, 8) graph with intersection array {343, 288, 1; 1, 96, 343} exist.



中文翻译:

对映Kerin图和接近它们的距离规则图

摘要

直径为3的对映非二分距离规则图Γ具有交集\(\ {k,(r-1){{c} _ {2}},\; 1; \; 1,\; {{c} _ {2}},\; k \} \)\({{c} _ {2}} <k-1 \))和特征值kn,–1和– m,其中n和– m是二次方程\({{x} ^ {2}}-({{a} _ {1}}-{{c} _ {2}})x-k = 0 \的根。的克林结合\(Q _ {{33}} ^ {3} \ geqslant 0 \)给出\(米\ leqslant {{N} ^ {2}} \)如果\(R \ NE 2 \)。在\(m = {{n} ^ {2}} \)的情况下,遵循Godsil,我们将Γ称为对映Kerin图。点图Σ展开的\(GQ(q,\; {{q} ^ {2}})\)给出了\(r = q + 1 \)的对映Kerin图。如果Σ具有固定分布的每个分量的f阶自同构σ ,则其顶点为σ的图\(\ bar {\ Sigma} = \ Sigma {\ text {/}} \ langle \ sigma \ rangle \)点集中的一个轨道和两个轨道是相邻的,如果一个轨道的顶点与另一个轨道的顶点相邻,则是距离相交图{ q 3\(((q + 1){\ text { /}} f-1)({{q} ^ {2}}-1)f,\; 1; \; 1,\;({{qq ^^ {2}}-1)f,\; { {q} ^ {3}} \} \)和每个局部子图Δ(u)对\(p {{G} _ {{f-1}}}}(q-1,\;(q + 1 )(f-1))\)。如果f= 2,则我们有一个\(GQ(q-1,\; q + 1)\)的伪几何图。因此,具有相交数组{ 125,96,1 ;的局部伪GQ(4,6)图。1,48,125}与本地伪GQ(6,8)与图形相交数组{343,288,1; 1,96,343}。

更新日期:2020-09-11
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