Classifying cubic symmetric graphs of order 88 <i>p</i> and 88 <i>p</i> ²

Abstract For a simple graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma is said to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s -regular, provided that the automorphism group of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma regularly acts on the set consisting of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s -arcs of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma . Given a positive integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n , the question on finding all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s -regular graphs of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n and degree 3 has received considerable attention. An <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>s</m:mi> </m:math> s -regular graph with degree 3 is so-called a cubic symmetric graph. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p be a prime. We show that if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma is a cubic symmetric graph of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>88</m:mn> <m:mi>p</m:mi> </m:math> 88p , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:mn>11</m:mn> <m:mo>,</m:mo> <m:mn>23</m:mn> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> p\in \left\{5,11,23\right\} ; if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma is a cubic symmetric graph of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>88</m:mn> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup>