# SageMathCell online https://sagecell.sagemath.org/?q=jhxqpt
#  (8,4) = 1100; Moore bound= 3201;
# Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006)
# Voltage graph Z_20 x(2) Z_55, B(0,4), voltages [(4, 27)(12, 11)(9,9)(19, 11)], avg. dist.: 3.547771
# Ord.: 1100 / Size: 4400   / Diam.: 4 / Avg.dist: 3.54777 
# 8-reg.? True / Degree histogram: [0, 0, 0, 0, 0, 0, 0, 0, 1100]  / Girth: 5
# Aut.group.ord.: 1100 /  Cayley ? True ---  vtx.trans. ? True -- edge.trans. ? False
#
# Com_1100  Cayley graph with 1100 nodes and 4400 edges non-isomorphic to the Loz
# Semidirect product Z_20 x(47)Z_55
# generators  [8,43]<>[12,42] : [11,8]<>[9,34] : [19,17]<>[1,26] :[12,54]<>[8,16] 
# Ord.: 1100 / Size: 4400   / Diam.: 4 / Avg.dist: 3.53685 
# 8-reg.? True / Degree histogram: [0, 0, 0, 0, 0, 0, 0, 0, 1100]  / Girth: 5
# Diameter: 4  avg. dist.: 3.536852 (1,8,56,373,662)  transmission:  3887  (lower than Loz)
# Aut.group.ord.: 1100 /  Cayley ? True ---  vtx.trans. ? True -- edge.trans. ? False
#

import networkx as nx

loz1100 = Graph(r":~?PK_?_?_?_?_?_?_?_?_I?H_H_H_H_H_H_H_H_R?Q_Q_Q_Q_Q_Q_Q_Q_[?Z_Z_A?Z_Z_Z_Z_Z_Z_d?c_c_c_c_c_L?c_c_c_m?l_l_l_A?l_l_X?l_l_l_v?u__?u_u_u_u_L?u_B?u_u_F_~_e?~_X?~_K?~_~_~_~_N`F_W@F__@F`F_G@F_q@F`F_S`N_O@N_^@N_e@N_|@N`N`N_]`B@V`V`V`V_U@V_D?d@V_q@V_i`^`^`^_b@^`H@^_I?p@^_|@^_r_T?{@f`B@f`f`f_C@f_k@f`Q@f_D?w_P@n_`@A@n`H@n`n_t@n`n_I@@`u_}@u`u_Y@u_g@I@u_A@Q@u_T@L`|`C@|_a@|_B?n@R@|_L@\@|`|_`@S_jACaC_K?x@]AC_X@dAC`JACaC_g@W_sAJaJ_F?W@eAJ__@gAJ`UAJaJ_n@c_yAQaQ_N?^@hAQ_e@pAQ`ZAQaQ_x@m`DAXaX_S?d@qAX_q@zAX`aAXaX`LaZA_a__FA__\?zA__gARA_a_aDA__G?m?y_m_e?m_`?m_m@T_m_maoAn`zABAnan_@An`mAn_{AHAn_hAn`TAn_G?j@v`_@v`vAa`Y@v`vAL`v`vb??wABA~`OA~_]@jA~a~a~ajA~a~_EA[A~_h@E`SBGbGbG`bAaBGbGbGaYbN_PA{BN_RBN`BBNajBNabBN`PBNae`qBVb@BVauBVaFAiBV_^@sBV_I?[BV_A?MBVaHAl`yB^a{B^b^afB^bLB^b^_|Apbe`M@yAVAmBe`pAzBeb@BebTBebe`AAybZBl_zBlblarBladB?Bla\BLBl`@@JBD_tBsbsbsaAAgBKBs_nBTBs_E?haqBSBy__BZBy`eBy_CBy`[AcBybhBy_pAdBPaOA|BYC@`ZABAHC@c@`EC@_fC@_JAgBXcFbUCF_WAvCFbBB`CF`gAaBhCFaqBccLb[CL_qA}CL`kAbBIBiCLajBqCLcLa|Bj_z@TBECS_C@]CSaiBQBrCS`DAuBwCSaMBaCSaZCS_UBBBm_i?tBMCZ_ICZ`WAeAtBxCZa{BzCZ_ABkCZ`JBIaYC`aKC`alAzB{C`_Y@iARB@CAC``[BoC`c`aBBQC?b\Cg`mANCg_yBLCJCgbvCgcgaIBc_DAUCiCm`DAeCmaxBRCmbBCaCm_[CmcTCm_nAYAfBH_@BH_NB@BHa|BHbHaKBH_v@X_vCJCR_O?v@D_vA`_vC?_q?vBSCX_g?vBCafBECG_DBtCGa@AICG`AAOCG_oCGC\cG`XCG``@}BPCR`}ASBf_O?s@}AyB}`}`i@}`}Cw`}bCB]_i@tBjDOdOdO`@AIDO`YDOdO_o@MCh`WDVaoDVbZDV`lCwDVcoDV_GCrcBD\_YDID\d@D\cVCvD\`[AzCDD\`HAaAkD\cXCx`{CIDcdEDc_d@wDc_T?fCsDcdTDc_L?RDcaOBTC{a?DjbdCICeCyDjdIDjd[Djdj`BBYDCd`Dpdp`TA?A]C}Dp`zCqDHDpcjDTDp`IBXBaDM`EDvdvcQCtDSDv_BBID[Dv`L@UBCaGC|DZD{_xA{D`D{_}BxD{acD{bpCpD{_M?oBKCqc_DFD_EA_eBoCRCXEA`hEA_PBAEA_{AhCtD^eF`MEFaVDKDeEF`aAIAOBzEF_VC|DheKdaEK_^BLDGEKb~CoDQDmEK`pCwDtEKdFDn_|BRDNEQacBrEQ`oCvDXDuEQb\D@DyEQc]DfEQciEQarDKDq`E@XBDDUEX_P@eAgEXbmCrD?DzEX`KDED|EXaTAaDoEX_bBaDQ_r?}ChE^_TC[E^`cCxDDD}E^avB|CaDIEBE^_LBpDrE^`UCRDX_GEdaRC?C^Ed_a@rAYDTEIEd`bDxEded_[?jARCYDh`RA]DBDYEj`qDKE_EjeREjbIChCsDP_YA`DPalDIDPdFDPaMBOBn_VAWBOEIamBOB\bOCn`U@x@~BObLBODZEVbBBODLaGCsDNEGadDwEG`\CPCYEG`TBYC_EGbJEGEZaPEG_t@Y@bBnEG_V?oBuCM_}CMCbDkamBMCMDCE?`ZCMaEB|CMcMEqaUCMdLDbaPBDCEDnFG`wFG`eFGbXCYFG`AFGfGbJBdEebmFNaNCzFNd`FN_EEqFN_X?[FNaDEC_C@XADAvFAaDEx_TADETEp_M?YADBpDD`g@tEV_eE}FYb?CHFYaqBAEmFYfLFYajAoFY_LC_D[EscOF_fAF_`zFSF_`oF_bZD_E{_wFWFd`_Fd_|COCkEuFd`PCJElF@Fd`iEgFLFdb`D^DfFEb]Fj_w@eFjePEnFKFjabDQFSFja]BcBkDLaKCWEtFRFo_^BQDEFWFo`O@rBUDzFocpFo_sAkBJDSEl_@?}@[E]E~FVFt_VB@DrEVFta?B{Ft`?BSCuEnFU_xBdFxceFCFZFxbvCYC_D|Fx`YAsEtF]f|`eFXF|_[E@FIFaF|`CCAEqFhF|e~FbbTDYFFGA`sCpDuGA_pEpFPFiGA_GExFmGAe[F[GA_CEfGA_nC}FCFeb]BnDMFMGH_j@{DqEwFnGHbbE}FpGHaICcFcGH_h@Z@wBUEe`wAqEY`J@t@wE|Fq`wAjDsFf_VBkFPafGQ_\@U@sCaE\GQaAA}CCChFLFzGQ`@@hFlGQgQ`ZAwBECaEWF]`lBiCkEzFQGW_OGBGWdQEeEmFH_IA\AvCnFH`KCxFAFHasCfDWFz_G?xCyDWaRDW_JBkCNDW_SDTDWFRGFdKDWFD`?@_CWEmFF`?CqFk`?BqEOEW_|@?D_E]_n@?DR_U@?@K`IAVAsBJEL_zBUELE`F`cyDUELE{FsboEL_ICUD~ELaFAPEL`_CdEL_U?yAEDMEEFbaECH`rAEBx`jAED^EWaEAZBYaEdRDiGR_r@xDqGvc^ErGvfWGv`LAPGv``AuAwGv`cCTFv`sCTGbaqCTGDG\_OBzCEGFb@GgH?`bDHEHH?_aGtH?`QCwCzH?ajE]FSG]aAENHDgjHDcJG{HD_p@~HDaVD`FVGebPG~HIbtHIbTENEhG_HIb|GTGtHI`HDeFUF[GnbPBxHNa@G@GZGsHNcoFIG{HN`RCkDhDoFDc[EUG^GzHR_KAzDXE}G~HR`MBfCCFnHR`S@ZBMDRFKaNA`BUBpGLGhG}HV_FAsDIFfGFHV`CCOD}HV_R?zBWDZEoGZecGlH@HZ_qDxEWE]FpHZ`AC~G^HBh]`TBxH]a]AwGqHFH]`kAPAVB[EBH]_\GhHG_dD[FQGoHbcDEkFiHbbKG\GxHMHb`zAfGbHPHbgJHAHbbIEuGlHJ`BDbFEGuHh`yBECKFeGaHQHhdgGgHSHhcYEaHHHh`M@_BCBoCHGR_Y@hCHC|GIbaCECHGfHTa[CHCwFgHK_kAsDoGx_w@CCsHq_`AxBkCDE_GKHqcQDGEDEeGtH[Hq_XBXB{HOHqboDADNE_GGHB_EDmEhGdGyHvamHcHv`kFIGRGYGp`tBbGjGp_bC~FOH[_iAfBQFO_PCaFOahDoEMFOapFLFOGzHg_SFCFOGmbWBtEUGYGobWDtG?GGbTBWFVGLaSBIBWFJ_KArBWBba^B`CeC~DRF}_f@MBRF}GNHE_i@RFMF}GeaMDrF}agESFrF}_UCVF}btEbF}_f?sAYArCUFEHGcUEHcCCUDzcUCiD_fJF^Hr_J?hFeIQ`WA^E\IQ``G~IQ_N?UBcIQbuD@DAIQ`JERHXcDERI@c|ERHeHzaSAmD|EEHg`@F@FyIZa}IPIZbhEqErIZcwGLG{H{`Q@YCQI^iGI^_F@OEIIVI^bKCNI^ceFWG}IB_oIYIcdwIc`YD[GUH}Icd~HtIPIc_z@kAHFZHAIK_oDzIh_[GqIVIhbiEhF]FcGm`mEYGEH|IUIkaiDDFPGgIYIkbdDkEDHQIk_^BjBoDUFJGs`MC^CnDsHlIEIXIoaeC~FAHKHgIoaoBRD]FRG[Hy`xASGPIIIr`OBLFlGGGLHSIr_B@[BYEvH|I\iuckDAINI`IuaxIEIab?FSGyILIx_MGXHMIxdSHzISIgIx`c@kCJCsI@Ix_`AKHjI[IxaFDQG_IIIdcIDNEJHJI?IjI~f\IDIlI~eWGOIbI~_gBdBtDLDrEHHravB{EHEtHidfEEEHICImaBEHEqHLIebFC~FcISbPB[EmJF_MA|DBDoGMHkJFePF?FwGRIPIsJFauD^D}IiJF_\DrEyFFGMI\`L@qFaGUITJKcyIyJK`{B~GqHrHxIMbDCsDXGw_aCuFcF~Gwc{GtGwIUI}apGlGwIJaTD]DwGEHxIL_\A^D]FhH`d[D]G}HlcbDQD]GraiC}D]Dg`XClDeEcEvFJH^bABdDYH^HnI_aNBDBiGuH^IBaWC]FfH^ctGCHUH^`dArETH^_}@`@lDwH^bABMChC}ESGnIa`aESFyaLEDESFn`CESEfFVgrHCJGaWAhBCHJJc`~BmClGKJc`hBuIYJcalArDhJc`IExEyJcbaGBIq_MGBJR__?fEtGBI{_R?aAKBXGi_P@_AKDGJb`p@xEqHlIV_qJYJnaeBfFzJhJndSEMJn_XEcG~IXJTa@BJJkJr`yFkJrfrJIJbJrbRB~CXH@I[J]`iBJFnJv`rAwINJhJvdmGUHBHHIJc?GIHfJOJgJycvE|GxIDJkJydiF`FwIjJy_CAzDnDrFMGraRBdE\FgJAJWJjJ}_dCrEvGjIeI}J}_yCzDYFTGzJN_J@C@bCbHpJ[K@_\BfDTHOHlIlK@`GAbBpD_G`JOJmehEyJ`JoKC``DBJWJpakHwIgKE_fFKJeJuKE`JB~EIEmJRKE`hA|C[J?JlKEcVFIH}J[Js_{FFF{IdJQJxKJ_OJVJzKJgGHoJqKJbBDiDwFDFfFyJG_sAAF[FyJUJ{aPCRFyIfJt_o@a@~DaGY`~AkDFEzFcHmJ@`~D@FUFqJw`_BcCBHFHuJfcKE@INJGJMJ_dMEmFPIRa}EoHHH_IR`KEsIPIRJgKIc{IIIRJ\ccFTFkHfJMJ^aVAxClFTIwfSF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com1100 = Graph(r":~?PK_?_?_?_?_?_?_?_?_@_@_@_@_@_@_@_A_A_A_A_A?J_A_A_B_B_B_B_B_B_B_C_C_C_C_C_C_C_D_D_D_D_D_D_D_E_E_E_E_E_E_E_F_F_F_F_F_F_F_G_G_G_G_G_G_G?u`A?y@@_^@@`@`@`@_x@@_S@@`@`A`A_b@A`A`A`A_K?[@B_T@B`B_t@B@J_g?r@B`B`C`C_u@C_K@C`C_W@C`C`D_H@D_O@D`D`D`D`D`E`E_r@E_o@E`E_Q@E`E`F`F`F`F`F`F_~@G`G_w@G`G`G`G_q@H`H`H`H`H`H_S@H`~@v@}`]@}_b@}`}`}`u@}`R@}_d@}`~`~`a@~`~`~`~_i@K@ZA?`SA?_Q?[A?`rA?AG`f@pA?a?a@a@_SA@_rA@`VA@a@aA_yAA`OAAaA_LAAaAaAaBaB`pAB`nAB`PABaB_gAB_}ACaCaC_jACaCaC`{AD_~AD`tADaD_nADaD_m@oAEaE_aAEaE_sAE`RAEayArAxaYAx_t@aAx_PAxax_UAqAxaOAx`cAx_aAyaya]Ayay_OAy_I?fAy`hAHAVAzaPAz`P@ZAzanAzBBabAlAzaza{a{_n@pA{aSA{a{`vA|aLA|a|_e@LA|a|a|a}a}alA}_~AjA}aMA}_XA}`fA}_j@zA~a~`iA~a~a~__AvB?`{B?apB?b?_Z@mB?b?`lAkB@b@``B@b@_L@qB@_zAOB@bqBjBpbRBp`rA]Bp_^Bp__Bp_k@TBiBpbJBpa_Bp``Bq_cBqbVBqbq_}@OBq_v@I@eBqacBCBPBrbKBr_{AMAVBrbfBrBzb[BdBr_uBrbsbs`mAlBsbNBs_O?YBs_YArBtbGBt_~Bt`dAIBtbt_NBtbubu_h@{BbBubHBu`WBu_MAbBu`iAuBv_iBv_a?|AeBvbv_h@^BnBw_lAvBwbhBwbw_H@YAiBwbw_[AhBcBxbx_TA\Bxbx`LAmBx`wBJBxcgC`Cf_hCJCf_KAnBVCf`]Cf`^Cf`jAQC_CfcBCfbXCfa\Cg`bCg_ICNCgcg`zALCg_`@sAFAaCgb\B{CHChcCCh_l@xBHBPChc\ChCp_SCh_gCiciaiBdCicFCi`XBjCj_{C?Cj_a@{Cja`BDCj_V?sCj`NCjck_fCk`gAvCYCkc@CkaTCk_c@MB[CkaeBmCl`hCl_V@`@yB^Cl_TCl_|@gAZCdCm_g@kBnCmc^Cm_mCm_yAUBaCm_vCm_k@ZB`Cncn_z@SBUCn_MCnaIBeCn_dCBCnd[?eDTDZ`gC~DZ_o@KBfCNDZaYDZaZDZafBLDSDZcxDZcPDZa^D[_T@IDBD[_~D[_cAuBGD[`_AoBABZD[cSCqC}D\_hD\`kAsC@CHD\_pDPD\Dc`RD\d]baCZD]c{D]_bC`D^_Z@xCuD^``AvD^bYB|D^`U@qD^aKD^d_`eD_adBnDMD_cvD__tD_`bAJCRD_b^CcD`acD`_q@UA\AtCUD``SD`_m@yAdBSDXDa_O@fAgCdDadRDa_J@lDa_\@vBOCXDa`sDa`jAVCWDbdb`wAPCMDb_j@MDbbDC[Db_R@cCxDbeM@dEFELadDpEL_|@nAHC\DBEL_LBREL_wBSEL_MB_CDEEELdkELdDELbWEM`SAFDtEM`{EM`bBmC?EM__A[BgByEMdGDdDoEN`gENagBkCvC}EN_xEBENEUaOENeOdmEO`aDTEP`YAsDhEPcQCrEPaRAmEPbFEPeQaaEQb]CdE?EQdiEQa^BEDFEQb\ER`oARBUBlDIERaPER`lAtB]CKEJES_U@OAbDXESeDES`JAhES`[ArCGDLES_]AoESafBPDKETeT_dBKDAET_J@iAJETb|DOET`QA_DkETe{?NA`EtEzb]EaEz`yAjBCDPDtEz`LCJEz`tCKEz`MCVCyEsEze]Ez`?DvEzaPBAEdE{_{AvE{a^CcCuE{`^BTC]CoE{dyEVE`E|adE|_UCaDiDoE|_N@uEpE|bJE|_RE}_]?lE_E}a]EFE~aUBkEZE~dEDeE~_jBMBeE~b~E~f?_\BZF?cTDXEmF?e[F?bWB}DxF?cSF@akBMCMCbD{F@_WBKF@ahBlCTD?ExFA`TALB[EJFA_TErFAaGB`FAaWBjC|D~FA`\BgFAb_CHD}FB_I@cCCDsFB`JAeBEFBcrEAFBaNBXE]FBfg@NBYFaFf_XCTFNFfatBbB{EBEdFf_`AIC~FfapD?FfaJDJDlF`FffJFf`|EfFf_zBKByFQFg`xBnFgbWDWDhFg_eAZCLDQFgeiFCFMFh`TDUE[E`Fh`NAqF]FhcBFh_i@QFi`\@kFLFi_~BVEtFjbOCaFGFj_YDwEWFj_T@iCEC[Fj__@[Fk_jFkdHEJFZFkfHFkcOCsEhFk_HDGFlbcCEDADVEjFl`VCCFlb`CbDHDqFdFmaQBGCRExFm`SF_FmbBCWFmbQC`DnElFmaXC]Fm_sCVC}EkFn_h@IA_EcFnaGB^B}FndeEoFnbICPFJFn_ZaKCQGKGP`WDHFyGP_QBlCYCqEpFQGP_g@_BDDpGPbED|E^GJGPfuGPawFSGP`wCCCoF|GQasCdGQcOEIEZGQ`dBSD@ECGQfVFoFxGRaQEGFHFMGRaKBiGGGRcxGR_uAXAgFwGS`{CNFaGTcGDUFsGT`XEgFDGT_Z@^AW_Z@i_ZDzExGD_ZDCDfFU_yDyGU_kCzDsEHFWGU_hASGU_]CWDVDzEbGNGVbLC?DFFdGVaPGIGVbzDKGV_|CIDTFYGV_PDQGV_f@qDJDoFXGW_H@gAFBXFPGWbBCUCsGW_nEWF\GWcADDFuGW_T?f@Y_pATDzGaGv_s@fA[B|EaGv_MB}FKGpGvg^Gv_cBoF~Gv_fBkDX_]?fDCEwFG_fA`CKDrEqg@GXG`Gw_XBLEuFxGwbFC_GmGwc|EGG\Gx_bFTFpGx_{@YAZBQ`YAe_w@YFdGj`YDuEXG?`vEiGy_o@jEcEvGAGy`gBNGy_w@\DKEHFOGtGzcDCuDxGNGz_vBKGoGzcpD}Gz_K@yEFGCGz_^ECGz_\@eAmD|E`GBG{_yAdBACPF{G{`mFDGFG{cwDvG^G{_r@qAbBTCrFN_r@MCsFvHP_rHA_r@bCeGe`eCaEJ`\@eDuFcFs_L@eBYD?F^_HDnEuH?HU`aF}GYHU`xAUBSCIaUB^`tAUGNHJaUEeFEGf_RArFVHV`nAfFPFbGgHV_nAdCFHV`tAXD}EvFzHSHW_VCyDhEhGtHW`sCCHOHWdcEkHW`KAtEtGiHW`]EqHW_M@[AaBeFMGhHX_i@vB]ByDDGcHXaiFpGlHX_}DjEfHAHX`pAJDfG_Hj_v@p`pA^DYHE_qAaDUEx_YAXAaEeGMG\`LAaCQDqGH_~A]GdG|_KAsBOCKbOCUbOFRFqHF_O@QBjG@HoajB_F{GLHGHo`mB]C{Ho_PApEkFbGbHmHp`UDlEZFUHSHp_hAoHiHpeUFXHp_WAHBlFaHIHpaYF^Hp`MAWBZC[FxHHHq`hArCTCoDvHCHqbaGYHLHq_v@zE\FSHq_[?pAlBEEXIA`sAlalBWEKH_`oBZEGFd_P@XBZFRGsH?aIBZDEEbGn`{BVHDHY`KBkCGD?_JCGDI_`CGGZH`bbCVGcGrHaIFaiCTDmIF_^BhFXGLHBIDIG_qARE^FGG?HmIG`gBgI@IG_NGBIG`VBCCbGKHcIG_VBRGHIG__AJBQDOG`HbIH_eAcBjDHEfH]IH_HCXG|HfIH`sAuFIF~IH_x@ZBdB}FEIW_HAoBd_OBdCOEyHw__AkEuGN__BDDwFOHNavCNH^HraHCaC|Dq_\@JC|D{_n@_C|G}cYDJHCHRHxI\baDHE_I\`oBMFKFsGfIDI]adC]IVI]`NGhI]aSB{DVGqHzI]`UCJGnI]`dB\C`DzFSHuI^_yDLHYH}I^aoBmFtGeI^`uAVCZCsFqIk_yBgCZ_q@OCZDCFeIN`?@^BcGt_R@^B|EgFzHhbnDBHvIIbCDUDnEb`[AGDnEj_I@mA[DnHZdMD|H]HlIOIocXDzFLIoakCEFvG\HFIZIpb]DQIjIp_{AKHHIpbNCqEHHQIPIpaRC~HNIp_wA`CSDTF~ILIqbgCcG]HEIqaqBPDNDfGZI~`vC]DN`oALDNDuGOIc_k@|AZHS`QAZCrFTGbI?cdDtIMI__|B{EGFO_|AWBBFW_|@IAiBTHs_ME?HuICIdJA_wDLFwJAbcCzG_H?H`ImJBcTECI}JB`xBFHbJBcFDdEvHkIeJBbMDpHhJB`tBYDGEFGeIaJCc]DWH@H_JCbiCHE@EXG}JPakBGE@EeGuIv`jAwBSHmaNBSDeF}HBIUdXEdIbIr`yCqEuFz`yBQBzGA`yAFBaCLIJ`MEmILIYIwJS_Q@tD~JS_mDHEqJO_mAsB~Hy_mC{EVFbIBIx_mCEEaI?apCQDyEtHEItJT_uDQEIHwJTc_C}EnFEHZJ`_JBcC?EnFRHTJHafBoCKIDbICKEWGdH\IieJFQIuJD`?AtDdGbatCICpGgaJFZIaIlJIJc_Y@PApElJc_V@lDzF^J__[@lBkCt`lDmFCGLIXJJ`lCzFNIUbhDEEiFaH_JFJddSDoF[FqHsJo_`?k@JCuF[HnJYb_CeD?IZ_\ExF|JGJU`|BlEVHB_KBlDcHG_iBEGDItJ?JZJq_l@XAMBhFYJq_w@UAhGHJn`ZAhCaDg_OAhE_FoGrJ[_oAhFyIic^DwFVGKHwJWJreEE`GEGZIJJ|`_@jAGDhGEIEJicVDYDqIm_g@[FdJXJeawCbFCH\`KCbEUHa_a@hB}GjJFJQJ~_b@kBHC^GCJ~`tARB`GnJ{aVB`DUEY`OB`FLGXHRJj_]@nB`GaI|dREgG@GqINJgK?_\A[AfBBEZGkI[Jv_qDJEKEb`fAWGNJhJs_NAHDVHx_]@`AcCsHJJWJa_]@aAgC@DRGiapBMCWHNKH_WBPCWEGFF_s@\AjCWJN`[BTB_BzFGHKInKC`oD|EyFOabBQGtJuK@`NBCEHIO`\A\B\D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loz1100nx =loz1100.networkx_graph()
com1100nx =com1100.networkx_graph()

# List of graphs to process
graphs = [('Loz1100  ', loz1100),('Com1100  ', com1100)]


def count_k_cycles(G, k):
    count = 0
    visited = set()
    def dfs(path, start, depth):
        nonlocal count
        current = path[-1]
        # Early exit if we?re going too deep
        if depth == k:
            if start in G.neighbors(current):
                # Normalize to avoid duplicates
                cycle = tuple(sorted(path))
                if cycle not in visited:
                    visited.add(cycle)
                    count += 1
            return
        for neighbor in G.neighbors(current):
            if neighbor not in path and neighbor >= start:
                dfs(path + [neighbor], start, depth + 1)
    for v in G.vertices():
        dfs([v], v, 1)
    return count   # each cycle counted twice (once forward, once reverse)

def algebraic_connectivity(G):
    """
    Compute the algebraic connectivity (Fiedler value) of a graph G.
    INPUT:
        - G: a SageMath Graph
    OUTPUT:
        - The second-smallest eigenvalue of the Laplacian matrix of G
    """
    L = G.laplacian_matrix()
    eigenvalues = L.eigenvalues()
    eigenvalues.sort()
    
    if len(eigenvalues) < 2:
        return 0  # Trivial case: empty or isolated vertex graph
    return eigenvalues[1]


def domination_number(G):
    """
    Compute the domination number of a graph G using MILP.
    INPUT:
        - G: a SageMath Graph
    OUTPUT:
        - The domination number (integer)
    """
    p = MixedIntegerLinearProgram(maximization=False)
    x = p.new_variable(binary=True)   
    # Objective: minimize the number of chosen vertices
    p.set_objective(sum(x[v] for v in G.vertices()))
    # Constraint: each vertex is dominated
    for v in G.vertices():
        p.add_constraint(x[v] + sum(x[u] for u in G.neighbors(v)) >= 1)
    return p.solve()


# Print properties for each graph in the list
print("\n Main properties of the graph\n")
for label, graph in graphs:
    print(f"{label} | Ord.: {graph.order()} / Size: {graph.size()}  "
          f" / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} \n"
          f" / 8-reg.? {graph.is_regular(k=8)} / Degree histogram: {nx.degree_histogram(loz1100nx)}  / Girth: {graph.girth()}\n ")
          # f" /  Alg.conn. {algebraic_connectivity(graph).n(digits=6)}") #/  Domin. number: {domination_number(graph)} ")

print("\n Symmetry properties of the graph\n")
for label, graph in graphs:
    print(f"{label} | Aut.group.ord.: {graph.automorphism_group().order()} /  Cayley ? {graph.is_cayley()} ---  vtx.trans. ? {graph.is_vertex_transitive()} -- edge.trans. ? {graph.is_edge_transitive()}" )

def check_all_isomorphisms(graph_list):
    n = len(graph_list)
    print("\n Isomorphism check of all pairs (a dot means non isomorphic):")
    for i in range(n):
        label_i, G_i = graph_list[i]
        for j in range(i + 1, n):
            label_j, G_j = graph_list[j]
            if G_i.is_isomorphic(G_j):
                print(f"\n{label_i} is isomorphic to {label_j}")
            else:
                print(".", end="")
    print()  # Newline at the end

check_all_isomorphisms(graphs)    
    
# Compute the distance distribution from a given vertex v in graph G
# Returns a list where the i-th element is the number of vertices at distance i from v
def distance_distribution(G, v):
    from collections import Counter    
    distances = G.shortest_path_lengths(v)
    distribution = Counter(distances.values())
    result = [distribution[d] for d in sorted(distribution)]    
    return result

print("\n")
for label, graph in graphs:
    print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}")
#    print(f"{label} distance distrib from vtx. 2: {distance_distribution(graph, 2)}")
#    print(f"{label} distance distrib from vtx. 4: {distance_distribution(graph, 4)}")
        
# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 6")
for label, graph in graphs:
    print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 7)))

print("\n")

##