▁them ▁have ▁chrom atic ▁number ▁ 3 , ▁diameter ▁ 4 ▁and ▁g irth ▁ 5 . ▁They ▁are ▁non - h amilton ian ▁but ▁are ▁hy po h amilton ian . ▁Both ▁have ▁book ▁thick ness ▁ 3 ▁and ▁queue ▁number ▁ 2 . ▁ ▁Al gebra ic ▁properties ▁The ▁autom orphism ▁group ▁of ▁the ▁first ▁Bl anu š a ▁sn ark ▁is ▁of ▁order ▁ 8 ▁and ▁is ▁is omorphic ▁to ▁the ▁Di hed ral ▁group ▁D 4 , ▁the ▁group ▁of ▁symmet ries ▁of ▁a ▁square . ▁ ▁The ▁autom orphism ▁group ▁of ▁the ▁second ▁Bl anu š a ▁sn ark ▁is ▁an ▁ab elian ▁group ▁of ▁order ▁ 4 ▁is omorphic ▁to ▁the ▁Klein ▁four - group , ▁the ▁direct ▁product ▁of ▁the ▁Cy clic ▁group ▁Z / 2 Z ▁with ▁itself . ▁ ▁The ▁characteristic ▁polynomial ▁of ▁the ▁first ▁and ▁the ▁second ▁Bl anu š a ▁sn ark ▁are ▁respectively ▁: ▁ ▁General ized ▁Bl anu š a ▁sn arks ▁There ▁exists ▁a ▁general isation ▁of ▁the ▁first ▁and ▁second ▁Bl anu š a ▁sn ark ▁in ▁two ▁infinite ▁families ▁of ▁sn arks ▁of ▁order ▁ 8 n + 1 0 ▁denoted ▁ ▁and ▁. ▁The ▁Bl anu š a ▁sn arks ▁are ▁the ▁smallest ▁members ▁those ▁two ▁infinite ▁families . ▁ ▁In ▁ 2 0 0 7 , ▁J . ▁Maz ák ▁proved ▁that ▁the ▁circular ▁chrom atic ▁index ▁of ▁the ▁type ▁ 1 ▁generalized ▁Bl anu š a ▁sn arks ▁ ▁equals ▁. ▁ ▁In ▁