# SageMathCell online https://sagecell.sagemath.org/?q=lpnunq
# Moore bound=2042. 
# 
'''
dhar859    | Ord.: 859 / Size: 5329 / Diam.: 3 / Avg.dist: 2.83367 / 13-reg.? False / Girth: 3 
pelkh856    | Ord.: 856 / Size: 5564 / Diam.: 3 / Avg.dist: 2.81882 / 13-reg.? True / Girth: 3 
Degree histogram dhar859 : [0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 63, 59, 70, 634]
Degree histogram pelkh856: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 856]

Number of k-cycles for k=3 up to 4
dhar859     66| 11557
pelkh856     107| 11988

'''
import networkx as nx

dhar859  = Graph(r":~?LZ_@???K??@O??F???G@GB_?_@?E?K?W?o@_B?E?K?g@OA_D?I?S?g@OB_F?KBC?w@_G_F?M?[@GAOC_H?Q?c@GAOD_J?U?k@WAoD_J?Y?s@_XOE_L?Y?s@gBoF_N?]?{@wBoF_O?}@?LWC?^_P?a@CA?]OG@p?gAkAWCoH_R?e@KAWDOI_SC}@SAgDOI_T?mACAogOJ_UDQ@[AwDoK_WDE@_RGEOK_X?q@_VwE`V_Z?sHsBWEoLB@?u@gWWFOMB^?wHSB_rOMAr?y@oTWFONA|?{KcBodoNBb?}@wZWEoNAd@AA?\gG@L__DaA?YgG@f__DqA?^WG`H_aGMAG[wG`R_aFmAGYwG`b_aDmAO_PCoQBN@WCORWH@[_cFuAO\gH@W_cEiAWWgH`U_eC{QkCosoRCGGuAW_GH`\_eF]A_^@MOSCCGyA_Q`KOSAj@ONCD?voSBL@OJKDOyoTBqHeAgS@HOTBP@SLohwI`Y_iG[P{DOiOUA|@WLghGJ@JCn@WNwegE?UAd@WK_kGJ@p_kG_PcDojav_mEAAwR`JOVBd@[KGiWJaDCV@[L_hWJ`}D@@_M?iGK@sDt@_H_dGKAEC`@_Jw_WK@PDb@_NOfwK@dD\@cHGeWKaBDl@cNWcgL`ZDN@cMWjGK`jEP@cKWnWK`RC^@gIGaWL@sDV@gH_ewL@dDv@gNOcwL@oEF@gJwgpdOXCKJMBWU`PBH_uFcQKFogA\_uDOPSEpBawEf@kMgjwL`mEH@kL?ngM@cDx@{LgogM@JD?L}B_a@ZbU_wFCTKF?nAdE\@oIOawM@~Cl@sOglPio\B?IOX[FOfAYEt@sNodwM`kEKMaBgTpEO\BAJeBg[`WO]BkIyBoXp\O]AYH{[CF_tBDFB@wJ?cGNA@DaLQBo^PIBn_{DoT?rwN`nEMM]Bw_`YBP_}CgRovGN`XDALABwX`_Bt_}FoQgzgN`wDT@wEgTPEo_C?H_]CG?jAMF~A?Hwf@nPvDZA?Jgh`eo_BCJ{]KG@AAqEdA?LGoPqo`AQIC[KGO|bFE^ACMWbPiO`CEK?_sGOtaTF@ACJWf`vo`AeIg]SGOpasFzAGMGaPhOaB}KKXkG`CAzGHAGHWgpqoaBGJK^{G_hAjFbAGJoepwOaBYHKZpBgP`yECKeCW``]CB`EESUw}wP`KDIMMCWSPSbr`ED{S?yqJObB_HWZXBwP`sCULMC_Z@IB[GRAOMOc@gocB{KSYCR`SAkFNAOOgma@OcBAJC_CH?jamFiPqC_W@NbmGhASJgfPucT`IEcQGvqGOdBmGgYsHOfaaFCQYCg_@aBM`IGOVpAGQ`aDoNeCgT`VbsGvAWKol`~Cm`KDST_|qLOeAqHc\`CwR@IDMM_ckH_vaQEsOqCo]@FbT`KFoX?qwRaADyOECw`p^cFHnA[L?k`|Cl`MDOTg|aKofA_IW[`GwR`YCwMcaSHovAWEyOiCw\PFBW`KFcWGqgS@WD_N_cCI?mAYFSPED?Y`JbXGVA_Hgh@rCa`OFWOGb@jogByKWX`NwSA@DqOSesI?ratFwQmDGQPSbhHRAcLWlpdC]`QGKWO}aJohBuJoZhOgS`RDIL_`sIOpaUGMQaDG[pOBuHzAcJWapqCr`SDcPGwaXOiBoHk]`MwT@eCeO_cxRGT@IDUMseCI_varE[PuDOTPPbUG^AgOOop{cU`SFoVWvpuD@`UEgUOqQLojByJ{ZXQAgojAoIWYXCGT`LDYMkd{IozA[FeRiDWOOmAIC[MKeKIoraWGEQWiCIp?b@FuPUD_T@QBSGRAoLolPfc[`WG[Wou@}cS`WD?U?zqUOkAsGo[HLAoOkBOH[_pHAdokBqJc[@PAjokBiHg]PNGU`dCiOSchQWU`^CYMWf@VGU`yE?LcgxTGU`KDSMgdsJOga`EmOqDg``bb{GeUQDg\@NBwHrAsM?l@eCY`[FwVovabDU`[ECQH@APdL`[EoV?rQKomA[I{]@JQuomCIKO^xCaoomBcHs]HOGV@_CSMSexUwV@VDCLGZ@Aa[OnBmH{]xNwVaCEINwaHWgV`\C_M_epUWV`ND[ModX[GVa?DsLgghTWG_nBQJCXXGGV`UDOLS`X[wV`aCgOCcPQgWAGEOO?b@YGW@QDMLC`h\gT@pCqNSfsK?dakF[Qgl{K?qAQGCQCh{K?~a|EwSKiSK?uauE_P{nKK?nANFMRSjkKOcapFaRCmKKOpagE{PSnkKOhaSFIRgl[KOxbDF_P?jSKOlayEcQ{osKOtamF}S_hcKP@a^EYQWjCKO|aOGGPkk{K_{aHGCPgkSK_lAzEaQwo{\P`bmGWTeEOS@XbtHoVMEOZ`Tc?IMSiEOY@Pb[GeVyEOT@HbfHqV?pcK`BaZE_QSikKp@AYEQQKi[KovaoFuSWhsKo}AKGEPcksKodAtFeR[mSKo{BEFUOgjxcgX`TCmMKg@YREowAqJcYXJRAorBKIOZXCb?OsBgKO\`AQmeT`gDCQGyA]DnK\BOM?jp~DCI]XME_R@[BvHgVYE_Vp^BTHWWKrSL@BA\EUQOiSL?qaaEsPOnKL?|AIGAPolCLOualFySShHhgY`cDKM?aH^GY`]D}LOdX`BKotAUJG]PLayOtAcH_[HNquEN`iF{Pp@aODccOHoXxGaiOtBaKC\xAqmEW`kDWQoxa^DmKRBWO?aqAc^JFBWKOhPkcWJtBWHwlp{CtJjBWO_g@eC`ImYqEoYPSb{IESkscL_zbAFSOsk@dGZ@\DwL[dP_RMOvBMI[ZxDq~ovBkK_\hBaleP`mGCRGraSDUL\B[Hgl`yCrJoZIEw^PFcGG{UEEwV@]bUHQWGr[LotAkFqSGhhiGZ`WCcMGfhZRGOwB?K?Z@KBCE_`oGSRoqaRdPLTB_IwdPqCzJSWiF?R`YBuHcV[u[@_oa`EILsap]w[@qEMMc`xVbJOwB{Gs_xFQqex`oEoTO}a_dOLNBcHGmP{cxJqZeFGWpLBaIMVWqsMOlatGOOoiPhw|aoEkQSjhkw[`REKL{f@RrTOxBUGo]`Cqve_`qGKS_zQLEBKTBcMWdpfCmJO[UFOYPCbqGcUgrSM_marGKOgiHgw\`?CGIK\xERCEM`sC{VW~Q_DzL~BgKOepodEJmXUFOT``b_HmT?tKM_zaRE]QskhqG\A?DULOcXVbZFN`uFGQOqQTdbMFBkL_bpxcPJ[XyFWWPKbbIIVSq{MofA|FuR{nPoG\`VEOLkepQrUozB?JW_XAqjdbLDBkNoj@hcaJ?ZCx[MpAafFUQ?oHarhO{AwJO_`AQiedMzD_WwvaYDILVBoNgi`hCcIqZ_xkN?eb?F}Ronpng]A@DAN?b``rEfU`wE[RoxQaDrKdBoLObPwcSJ_X}F_\`IbLHWUGwKNOibDEyRShHjbrO|AAEWRwxA`dMJgXEFg]@IBIHSUOw{NOdAyFsRwn`nW]`{D[L_cHVBXFI`yGGT?yQNeCK_\UFgUPWcFGYTOt@uw]`nC_NWb@ZBLo}CKIW\pFbBEHMnBwNOjPgcgIsZOyCN_naqGCOwj@hBko}AWJ{_@NQ}e{`{F?Po|QJdjKxBwM_e@gChJM[QFoXPMBfICVCq`yw^@PEALoeXRBUFg`}F{TwtqRd^Lm[eFw[PGbKH_UWwp|G^`QEGLcePRbWFc`}CkVo~A\e?LtB{JomAAcNIeYCzcNouaIFoPWlHfW^aGDCMgbh`REFS`}EORgxa_dqKe]qG?Y@OBgIOV_r@{G_AFDIMobX_bFfPa?EwPW|qIdMJYXeG?S@]B}HsV{vX~g_@tCkKkdxWR`Fta?FcTGtQRDZLi[yG?T@`BYHaSstxxw_@YDmOCa@TbRFYaACcWH?Q_e@MA_EGGUpKB}GyVStXqg_`RD}KkapWBQfiaAGKTgyAFdyK}^]GGWpFBkHuT[v@xg_`jCwLSePWrEg?aAFkSo|qWDJKc\mC?OOj@rDcLgaxHqvE}MjCGKGaPtCyIcZG{SO_~AbFoQ{hxcRmpABcJKZHHave|Mk_qGOR``cDIOWKwy@w`@_CeNobX\bVFLaCD[VWrAId\LO]eGOZ@LbVHaU_q@}xAajFKP?ax]RLfwaEDGVOqQHdZLE]mGWa@VBbGWV{rP|cGpBB}I_]XJaeeTMrCKHWpQ@dFKC[`?kOoua^EeR_kHbb|PBA{HO^XEa{EhM`CKMGkPlcdJYZ{yyBW``cC_MkfHTbZFgaGGOT_xACd|Ky^TAkP?zauEyQOlXmbggIaGEcRGuAYDfKW^yG_RpcCFIGWWwq?wa@UDyKga`UbOfoaGEGPwyQ]DWLm]\BsP?maQFsPomHkBfPCC?I[]pJAeERMtCSNghPyclIQXWzsPOtA_EaR[k`ab}pDAwHW_@EQyEiMUaMGgRP`CAIKWOwiAGa`fCUMwg@TBXf`OzCSOGiPrcIJwY?}YDwa`WDwL?aPVBRfnaIFWUwvAPdkLq\H@sP_vA]EkRWkpaR~gnaGCWI_o@fCPI}YO|sP_sALFYR{iXlBqgXaKD?WxAAaeDMG_IGo]PObsHUSwq`wGb@tDgL{cPYR]FROVCWOwi`pCJK?Xo}IDgb@YCiNcc@[bUFNPJC[Jwdp~c^JmY{xIIGb`dCWN?fpSRYfcOvC[Ook@qcLJuXc}aCgb`KEKOOgXaB`GCaMFOUgvaOdiLu\P@{PowAYEcRSkhbr}GhaMFgS_{QTDOKo\qGwSP[bLGoTwspzCYpGBoJ_[@IAwF?Mo_eH?UPGb|G{VKthrCQPGAiJwXxDqkeaNYbQHOp@kCEICW[wY@gcAAD]MC`p^bNFqOhC_N_g`xChISX[zyNGc@eCSM{fhTRWfbOxC_LwfPiCrJCWk~IJG??E?M?a?IQW@HH_EcdHS@CchIQV?THIQseO??KXLHK@scxHaQCgHWR?e`MA]D@HSSOh@RAgDSIwUCchIaqDgJWV?m`]A}E@HST_o`aBEEOKoXosHIbIEcLOYou`mB]F@HS[Ox@rBgFSMo\o{HIbqFgNW^?}`}B}G@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pelkh856 = 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@SCgPOihuGpNM?]hOyaTDIJSWgtPqbuGKPa][lNiaDGI\S{h|SAgTPica]``Ji]tGieTOidTQitUIlT\jHYItDijUUml`ZIutqJwVvLT?QmD[iyTujpVindvkZY`DcNhOtdJJUWktXqrtgJ[WhIkoivTnJ^V?mD[QwuCKVX@?t]I~D}j}V}o@_J?e@kDWL?_AwUBQf?UQmXfzSehlSYitZ~~~~~~~")
             
dhar859nx = dhar859.networkx_graph()     
pelkh856nx = pelkh856.networkx_graph()

# List of graphs to process
graphs = [('dhar859   ', dhar859 ),('pelkh856   ', pelkh856 )]



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()} / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} / 13-reg.? {graph.is_regular(k=13)} / Girth: {graph.girth()} ")# /  Alg.conn. {algebraic_connectivity(graph).n(digits=6)}  /  Domin. number: {domination_number(graph)} ")

print("Degree histogram dhar859 :", nx.degree_histogram(dhar859nx))  
print("Degree histogram pelkh856:", nx.degree_histogram(pelkh856nx))  

           
           
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()}" )
print("\n") 

  
    # automorphism group structure
print(" Automorphism group structure: x means direct product; : means semidirect product.\n") 
Adhar859  = dhar859.automorphism_group()
print('dhar859 :', Adhar859.structure_description(), ' |  center order:',Adhar859.center().order())           
Apelkh856  = pelkh856.automorphism_group()
print('pelkh856:', Apelkh856.structure_description(), ' |  center order:',Apelkh856.center().order())           

print("\n") 
    
'''    
print("\n Properties of the graphs as at arXiv\n")
for label, graph in nonisographs:
    print(f"{label} & {graph.average_distance().n(digits=6)}  & {graph.girth()}  &  {algebraic_connectivity(graph).n(digits=6)}   &  {graph.automorphism_group().order()} \\\\ ")
''' 


# 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


for label, graph in graphs:
    print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}")
    print(f"{label} distance distrib from vtx. 9: {distance_distribution(graph, 9)}")
    
# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 4")
for label, graph in graphs:
    print(f"{label} ", "| ".join(str(count_k_cycles(graph, k)) for k in range(3, 5)))

print("\n")

'''
# from 0 versio Molodtsov 
for name, G in  graphs:
    fname = name.strip() + "_adjlst.txt"
    s = "; ".join(
        "{}-{}".format(v, ",".join(map(str, G.neighbors(v))))
        if G.neighbors(v) else str(v)
        for v in G.vertices(sort=True)
    )
    with open(fname, "w") as f:
        f.write(s)


# from 0
       
for label, _ in  graphs:
    base = label.strip()            
    G = globals()[base + "nx"]      

    with open(f"{base}_edges.txt", "w") as f:
        f.write(",".join(
            f"{{{u},{v}}}"
            for u, v in sorted((min(u, v), max(u, v)) for u, v in G.edges())
        ))

'''
# from 1
        
for label, _ in graphs:
    base = label.strip()   
    G = globals()[base + "nx"] 

    with open(f"{base}_Zedges.txt", "w") as f:
        f.write(",".join(
            f"{{{u+1},{v+1}}}"
            for u, v in sorted((min(u, v), max(u, v)) for u, v in G.edges())
        ))
       
print("\n DONE \n")  

##