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On bipartite tetravalent graphs
CS 10 W’Dr) 16747’2.0
North-Holland Publishing Company
Received i 5 March 1%74 *
et. in this paper tuedetcrnrination of all distance-transitive graphs of valexy
four is com-
In [ IO, 1 I 1 all non-bipartite distance-transitive graphs of vallency four have been determined. e use a result of Gardinrzr [ 4 ] to enable us to determine thf: bipar ite distance-transitive graphs of valency four. We wx tht definitio s and notation o Hwish to exp C’ollege, withou ould not have been completed.
ide .
3. iir feasible intersection matrices \Vt2now ~olk~~ the method of [ 11 1and use a computer ta ex
matrices of size up to 25~ 25 with integer entries in e computer checks whether
his is the intersection arra:,r sf t:le ~npletc: bipartite graph K4 4. K4. 4 is tht: unique distance-transitive grapt v.rith this i,ntersection arriy.
is the intersection array of the graph with vertices (li (i ::=1 5) and w-th edges ai bj # j, i, j = 1) 2, . . . . 5 11.T e-transitive graph with is array as intersecti& array.
169
This is the intersection array of the &cage of valency 4 which is the Levi G(2, 3). Thir; i:xthe unique distance-transi ive graph vvifh this array as intersection G”ray [ 7 1.
This is the intmsection array of the kube, which is th;; unique distancetransitive graph with this ma;): as intersection army.
We will show that there cannot exist a distance-transi;ivc graph with this array as intersection array. Suppose such a graph exists. Then the girth is four and given two vertices x and y such th;lt L&X,y) = 2, x and y lie on precisely one circuit of length four. Since each vertex of r2 is adjacent to two vertices of rt, we can begin by constructing miquely as shown in Fig. 1.
v;t) = 2, we cm join ~2 to yvi to mq~kt~ a circuit of length 4. Sir;&&, we Suppose ~2 ‘IB joined a 1;ir, k-3.Since d(us ar2)= 2, we can join iete a circuit of length ,3 imd tr2. Then &t~, ~4) := 2 ttrt we cannot a circuit e,f length containing ~1 a~ d ua. I 4 P’PIC~ a dis ce-tramitive grapl~i with this on may cannot exist,
be
an &cape of is nst dista: ~Hransitive [ 2 1. orkr as yet ~~~p~~~is~le~ that there alre *-IQ PQbe distsrm-transit ive,
ipffrtitr tetrrrvaht graphs
ces
of these
171
would
) = (u: d(u, u) = i in I’*}
this array as intersection array must be The 12-cane of valency 4 (see [ I 1) is kno;JIIn to es not-appear to be wn whether tJlis is mce-transitive 1 I-cage of valenc
I\ as shown in 191 that a uniq distance-transitive graph exists \vith te graph is a double cove array as interse&m array. graph with 35 vertices described in [ I If), i.e., ars antipodal graph ulrith (see [$I).
D.N. Smith, Highly syr imetrical graphs of low v&ncy, Ph.D. Thesis, University of Southampton, 193 1. . Smith, On tctsavalent graphq J. London 31th.Sot. 6 (2) (1973) 659-652. ath. Sac,, to appear. . Smith, Dis:tance4ransitive graphs of vaberacy four, J. London
North-Holland Publishing Company
Received i 5 March 1%74 *
et. in this paper tuedetcrnrination of all distance-transitive graphs of valexy
four is com-
In [ IO, 1 I 1 all non-bipartite distance-transitive graphs of vallency four have been determined. e use a result of Gardinrzr [ 4 ] to enable us to determine thf: bipar ite distance-transitive graphs of valency four. We wx tht definitio s and notation o Hwish to exp C’ollege, withou ould not have been completed.
ide .
3. iir feasible intersection matrices \Vt2now ~olk~~ the method of [ 11 1and use a computer ta ex
matrices of size up to 25~ 25 with integer entries in e computer checks whether
his is the intersection arra:,r sf t:le ~npletc: bipartite graph K4 4. K4. 4 is tht: unique distance-transitive grapt v.rith this i,ntersection arriy.
is the intersection array of the graph with vertices (li (i ::=1 5) and w-th edges ai bj # j, i, j = 1) 2, . . . . 5 11.T e-transitive graph with is array as intersecti& array.
169
This is the intersection array of the &cage of valency 4 which is the Levi G(2, 3). Thir; i:xthe unique distance-transi ive graph vvifh this array as intersection G”ray [ 7 1.
This is the intmsection array of the kube, which is th;; unique distancetransitive graph with this ma;): as intersection army.
We will show that there cannot exist a distance-transi;ivc graph with this array as intersection array. Suppose such a graph exists. Then the girth is four and given two vertices x and y such th;lt L&X,y) = 2, x and y lie on precisely one circuit of length four. Since each vertex of r2 is adjacent to two vertices of rt, we can begin by constructing miquely as shown in Fig. 1.
v;t) = 2, we cm join ~2 to yvi to mq~kt~ a circuit of length 4. Sir;&&, we Suppose ~2 ‘IB joined a 1;ir, k-3.Since d(us ar2)= 2, we can join iete a circuit of length ,3 imd tr2. Then &t~, ~4) := 2 ttrt we cannot a circuit e,f length containing ~1 a~ d ua. I 4 P’PIC~ a dis ce-tramitive grapl~i with this on may cannot exist,
be
an &cape of is nst dista: ~Hransitive [ 2 1. orkr as yet ~~~p~~~is~le~ that there alre *-IQ PQbe distsrm-transit ive,
ipffrtitr tetrrrvaht graphs
ces
of these
171
would
) = (u: d(u, u) = i in I’*}
this array as intersection array must be The 12-cane of valency 4 (see [ I 1) is kno;JIIn to es not-appear to be wn whether tJlis is mce-transitive 1 I-cage of valenc
I\ as shown in 191 that a uniq distance-transitive graph exists \vith te graph is a double cove array as interse&m array. graph with 35 vertices described in [ I If), i.e., ars antipodal graph ulrith (see [$I).
D.N. Smith, Highly syr imetrical graphs of low v&ncy, Ph.D. Thesis, University of Southampton, 193 1. . Smith, On tctsavalent graphq J. London 31th.Sot. 6 (2) (1973) 659-652. ath. Sac,, to appear. . Smith, Dis:tance4ransitive graphs of vaberacy four, J. London