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：

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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k，n，–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}。