# SageMathCell online -> https://sagecell.sagemath.org/?q=idmkwp
# 
'''
 Main properties of the graph
rajiv1518  | Ord.: 1518 / Size: 4554 / Diam.: 5 / Avg.dist: 4.32762 / 6-reg.? True /  Girth: 6   
loz1404    | Ord.: 1404 / Size: 4212 / Diam.: 5 / Avg.dist: 4.27655 / 6-reg.? True /  Girth: 6   
 Symmetry properties of the graph
rajiv1518  | Aut.group.ord.: 6072 /  Cayley ? False ---  vtx.trans. ? True -- edge.trans. ? False
loz1404    | Aut.group.ord.: 702 /  Cayley ? False ---  vtx.trans. ? False -- edge.trans. ? False
Degree histogram rajiv1518 : [0, 0, 0, 0, 0, 0, 1518]
Degree histogram loz1404   : [0, 0, 0, 0, 0, 0, 1404]

 Automorphism group structure
 C_n is the cyclic group of order n;  x means direct product; : means semidirect product.
rajiv1518 : PSL(2,23)  |  center order: 1
loz1404   : C2 x (C13 : (C9 : C3))  |  center order: 6

rajiv1518  distance distrib from vtx. 0: [1, 6, 30, 149, 608, 724]
loz1404    distance distrib from vtx. 0: [1, 6, 30, 149, 604, 614]
'''

import networkx as nx

loz1404 = Graph(r":~?T{_?_?_?_?_?_?_G?F_F_F_F_F_F_N?M_M_M_M_M_M_U?T_T_T_T_T_T_\?[_[_[_[_[_[_c?b_b_b_b_b_b_j?i_i_i_i_i_i_q?p_p_p_p_p_p_x?w_w_w_w_w_w`??~_~_~_~_~_~`F@E`E`E`E`E`E`M@L`L`L`L`L`L`T@S`S`S`S`S`S`[@Z`Z`Z`Z`Z`Z`b@a`a`a`a`a`a`i@h`h`h`h`h`h`p@o`o`o`o`o`o`w@v`v`v`v`v`v`~@}`}`}`}`}`}aEADaDaDaDaDaDaLAKaK_SAKaKaKaKaSARaRaRaRaRaRaZAYaYaYaYaYaYaaA`a`_hA`a`a`a`ahAgagagag_GAg_PAgao?KAnananananan_@_@?}_@_@?N_@?]a{Azazazaz_\Az_eAzbB?`BAbAbAbAbAbA_U_U@R_U_U?c_H?UbNBMbMbMbM_qBM_zBMbU?uBTbTbTbTbTbT_j_j@g_j_j?x_j@GbaB`b`b`b``FB``OB`bh@JBgbgbgbgbgbg`?`?@|`?`?@M_r@?btBsbs_GBsbs`[Bs`dBsb{?_@_Bzbzbzbzbzbz`T`TAQ_H?c@T`T@b`T@qcGCFcF_\CFcF`pCF`yCFcN?J@tCMcMcMcMcMcM`i`iAf_N?]@i`i@w`\@icZCYcY_qCYcYaECYaNCYca@IAIC`c`c`c`c`c``~`~Ay_r@M@~`~AL`~A[cmClcl`FClclaZClacClct?tA^CscscscscscsaSaSBL_x@GASaSAaaFAS_Kd?d?`[D?d?avD?dF@sAsDEdEdEdEdE_PDEahahB_`\@wAh_QAhAuahBC_`dQdQ`pDQdQbIDQdX@^BFDWdWdWdWdW_eDWa{a{Br`b@qA{_fA{BHapA{_udcdcaEDcdcb\DcdjA]BYDididididi_zDibNbNCEaFAaBN_{BNB[bNBi`JduduaZDuduboDud|AHBlD{d{d{d{d{`OD{babaCXaLA[Ba`PBaBnbVBa`_eGeGaoEGeGcBEGeNBEC?EMeMeM_CEM__EM`dEMbtbtCkapBHBt`eBtCAbtCO`teYeYbBEYeYcUEYe`ArCRE_e_e__XE__JE_`yE_cGcGC~auBCCG`zCGCTb|CGaIekekbUEk_DEkchEk_SbkCeEqeqeq_mEq`IEqcZcZDP_WBVBnCZaOCZCgcZCua^e|e|bhE|_YE|c{E|_hbXCxFBfBfB`BFB_tFBcmcmDb_BB[BiCmadCmCzcbCmasfMfMb{FM_nFMdMFM_}cQDJFSfS_AFS_N@WFS`sFS_ZD@d@Dt_A@AB|CTD@awD@DL_QD@DYbFf^f^cNF^`CF^d_F^`Rb~D\Fdfd_VFd_c@lFd`^Fd_EDRdREF_V?lCACODRbJDRD^_fDGDRbYfofocaFo`XFodqFo`gcwDnFufu_kFu_xAAFua]Fu`DDd_IDdEX_k@kCbCzDdb]DdDp_{DdD}blg@g@ctG@`mG@eCG@`|cdE@GFgF`@GF`MAVGFaHGF_oDv_^DvEj`@@VCgCuDvbpDvEB`PDkDvc?gQgQdFGQ_ZABGQeUGQaQd[ERGWgW`UGW`bAkGWbEGW`nEH_sEHE{`UAUDGD^EH_XCCEHET`eEHEacRgbgbdXGb_EAWGbegGbafdIEdGhgh`jGh`wA~GharGh`YEZ`HEZFL`jA@DLDYEZ_CCVEZEf`zEOEZcegs_XGsdjGs`DAlGsexGsaye?EvGygya?GyaLBQGybkGyaXEl_V@]ElF]a?A}DkEBEl`BCiElEwaOElFDcxhD_CHDd|HD_oB?HDfIHDbLdmFGHJhJaTHJaaBdHJbXHJaCE}_A@rE}FnaTAjDpD}E}_mC|E}FHadEsE}dJhU`BHUeNHU`nBRHUfZHUb_ecFXH[h[aiH[auBwH[cQH[b@FN`@AGFNG?aiBcEOEfFN`lDNFNFYawFNFfd\hf_mHfe`Hf`YBeHffkHfbreQFiHlhla|HlbHCJHlb~HlamF__kA\F_GPa|BPETEaF_`WD`F_FjbJFUF__IDnhw`lHwerHwaXBxHwcE_RFFFzH|h|bOH|b[C]H|cwH|bfFp`jAqFpGa_YBOCIEsFHFpaVDrFpF{b]FpGH_^E@iG`WIGfCIGaCCKIGcX_gEuGKILiLbbILbnCpILcdILbSGA`UBDGAGr_DBbBvEwFDGAaAEDGAGLbpFwGA_sERiWaVIWfTIWb@C^IWck_|FhG\I\i\_aBuI\cADCI\d[I\cLGR_]ATBWGRHC`CBuCoFUFjGRa~EVGRG]cCGRGj`HEdigaAIgfeIgamCqIgc~`QFWGmIlil_LCHIlcTDUIldIIlbyGc_HA?BjGcHT_nCHC\FYFfGcakEhGcGncVGYGc`]Ev_WIwa~Iw_AFvIw_KBfIwdP`fGJG~I|_B?LI|`KC[I|cgDgI|e?I|crGt`GA|B}GtHe`mC[DTFwGLGtbdEyGtH?_dCiGtHL`rFG_BJGakJG_VGGJG_`BSJGdb`{FyHOJL_W?aJL_vCnJLczDyJLdmJLc_HE_rAiCPHEHv`XCnDBF{GHHEbQFJHEHP_OC|G{HEaGFX`AJWbdJW_R?kGXJW_c?uCLJWdtaPGlH`J\_l?vJ\`uDAJ\_dDLEKJ\ecJ\dVHV`qBbCcHVIFaWDADxGYGnHVcJF[HVHa`NDNHVHna\Fi_lJgbQJg_g@@GiJg_N@JByJgeFaeG[HqJl`A@KJl``DSJl_OD^E]JleQJldDHg`\BOCvHgIVaBDSDfG]GjHgbwFlHgHr_yD`H]HgaqFz_Y@kJwcJJw_|@UGzJw`M@_CrJweXaxHNIAJ|`V@`J|a_DeJ|`NDpEoJ|fFJ|_YDzHxa[CHDHHxIfb?DeE\G{HPHx_dCpF}HxIB`xDrHxINbDGK_D@VKGbwKG`Q@jHKKG_x@tC_KGejbKG}IQKL`k@uKLaJDwKL_yEBF@KLeuKL_DDhIHaFBuDZIHIvalDwEJH?HLIH_OC]GNIHIR`cEDH~IHbWG\`CAUKWcpKW`fA?H\KW`wAIDVKWe{b^HpIaK\a@AJK\bGEIK\`xETFQK\fhK\`CE^IXbCCnDlIXJFbeEIF?H]HrIX`NDUG_IXIbabEVIXInbjGm_nA@Kgc]Kg`{ATHmKg`bA^DDKgfLbqH_IqKlaUA_KlatE[Kl`cEfFbKlfWKl_nELIhapC[D~IhJVbRE[EnHaHnIh_yDCGpIhIraMEhI^Ihb}G~`mA}Kw_JDUKwaPAiH}KwaaAsDzKwf]cDIPJAK|ajAtK|bmEmK|abEwFsK|gJK|`mFAIxbiDSEPIxJfcKEmFaH~IRIx`xDyHAIxJBcPHO`XAjLF__DCLFaeA|IMLFaLBFDhLFfncWI@JQLKa}BGLKbZE~LKaMFHGDLKfyLK`XEpJHbVDAEbJHJvbxE~FPIBINJH`cDgHRJHJRccH`aWBcLU_tDyLU__AxBOI]LUbHBYE^LU_ZG?cjIpJaLZbPBZLZcSFOLZglLZaWFcJXcODwEtJXKFcqFOGCI^IrJXabE]HcJXJbcvHqaBBPLc`IDgLc_JBKBbImLcauBlELLc_EGPc}I`JqLhbcBmLhc@F`Lhg[LhaBFRJhb|DeFEJhKVc^F`FrIbInJhaMEKHtJhJr_\DHIA`^E]Lq`IB^BuI}LqbnC?FALq`DGadOJPKALubvC@LucyFqLuhNLub?GEJxcuE[FVJxKf_`FqGeI~JRJx_GDZIQ`sEKL}_tBqCHJML}b[CREpL}_oGrdaJ@KQMAcICSMAcfGBMAg}MAalFtKHcbEIFgKHKv_KGBGTJBJNKH`FDlIaaHF@MI`sCDC[J]MIcTCeFcMI`nHCdsJpKaMMc\CfMMd]GSMMhpMMbeGgKX_WDYE~FxKX`JGSHGJ^JrKX_qD~Iqa]EoMU`^CWCnJmMUcACxFRMU`YHTeEJ`KqMYcoCyMYdKGdMYh_MYbRGVKh_BDGEmGIKh_uGdGvJbJnKh`pEPJAarFbMaa]CjDAJ}MaczDJGEMaaXHeeWKPLAMedBDKMeeAGuMeiPMecKHIKx`AD}F`GZKx`tGuHiJ~KRKx`[EbJQbEFQMmaHC}DSKMMmcgD\FtMmaCHveiK@LPMqdTD]MqdoHFMqi@MqbxGxLG_lDkFOGkLG`_HFHXKBKNLGaZEtJabXGDMybEDODeK]Myd^DnGgMyb@IF_XEzKpL^M}_RDfDoM}eeHWM}ipM}cqHkLV`kEaGBG|LVa^HWIJK^KrLVaEFEJqbkFsNEarDaDwKmNEdLE@GVNEamIV_CFKK`LlNI_gDxEANIeSHhNIi`NIc^HZLd`VEOFqHMLdaIHhHzKbKnLdbBFVKA_fB~GfNQbkDsEIK}NQeBERHINQbfIf`BF\LOLyNU_J?|EJESNU_hHyNUjPNU_\AUFDGdH^_\BFHyIjK~LQaoFgKQ_QCQGUN\bXEEE[LLN\dpEdGxN\bSIv_mFmL@MEN`__@QE\EeN`_SIIN`j@N`_GA@EsGSHo_GAsIIIZLBLMbhFxKa`PCdHHNg_dCQEWEmL[Ng_HEfEvHkNgcLJF_E@lF~MQNk_S?t@fEnNk`RIYNk_LJpNk`FA}FfHFI?`FBlIYJJL\LmbUGIKq_{CwGwNr_OB~EiE~LiNr_]ETFGHZNrbyJV_Z@WGOM]Nv_h@I@{F?Nv_}IiNv_aJ`Nv_qAjFUGuIO_qBYIiIzL_LjcNGZLA_R@zDIHjN}`NCwEzFOLvN}_^?rFHFXN}crJf_oAVG`MiOA_}@^APFPOA`|IyOA_vKPOA`pBcGHHhI_`pCRIyJjLwMFb{GkLP_g@eD[HYOH_yCdFKF`MBOH_I@GEwFiOHc_Jv`DAAGqMuOL`R@sAeFaOL`gJIOL`KK@OL`[BPFwHWIo`[C?JIJZLzMCctG|L^_|AdDmIKOS`xD[F\FqMNOS`H@\FjFzOSdVKF`YA~HBNAOW`gAHAxFrOWafJYOW``KpOWaZCIGjIIJ?aZCxJYKJMOM^caHMLl`QAOE?H{O^`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rajiv1518 = 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loz1404nx =loz1404.networkx_graph()
rajiv1518nx =rajiv1518.networkx_graph()

# List of graphs to process
graphs = [('rajiv1518 ', rajiv1518),('loz1404   ', loz1404)]


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 check_all_isomorphisms(graph_list):
    n = len(graph_list)
    print("\n Isomorphism check of all pairs (a dot means the pair ARE 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(".", end="")
                print(f"{label_i} IS isomorphic to {label_j}")
            else:
                print(f"{label_i} NOT isomorphic to {label_j}")
                
def non_isomorphic(graph_list):
    reps = []
    labels = []

    for label, G in graph_list:
        if not any(G.is_isomorphic(H) for _, H in reps):
            reps.append((label, G))
            labels.append(label.strip())   # remove extra spaces if you want    
    return labels

def isomorphic(graph_list):
    labels = []

    for i, (label, G) in enumerate(graph_list):
        if any(i != j and G.is_isomorphic(H) for j, (_, H) in enumerate(graph_list)):
            labels.append(label.strip())

    return labels

def compare_graphs_list(graphs):
    canon = {}
    count =1
    for label, G in graphs:
        s6 = G.canonical_label().sparse6_string()

#        print(f"{label.strip():15} : {s6[:40]}...")

        canon.setdefault(s6, []).append(label.strip())
    print("\n noniso ->",len(canon))
    print("\nGroups:")
    for group in canon.values():
        if len(group) > 1:
            print(count,"Isomorphic :", ", ".join(group))
            count=count+1
        else:
            print(count,"Unique      :", group[0])
            count=count+1



print("\n START \n")

#noniso = non_isomorphic(graphs)
#print(len(noniso), " non-isomorphic -> ",noniso)

#iso=isomorphic(graphs)
#print(len(iso), " isomorphic -> ",iso,"\n")


compare_graphs_list(graphs)


nonisographs = graphs



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

print("\n Symmetry properties of the graph\n")
for label, graph in nonisographs:
    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("Degree histogram rajiv1518 :", nx.degree_histogram(rajiv1518nx) )  
print("Degree histogram loz1404   :", nx.degree_histogram(loz1404nx) )  
    
'''    
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()} \\\\ ")
''' 
    
# Check isomorphisms
# print(f"Are isomorphic G5 and G6? {G5.is_isomorphic(G6)}")

#check_all_isomorphisms(nonisographs)

print("\n")


# automorphism group structure
print(" Automorphism group structure")
print(' C_n is the cyclic group of order n;  x means direct product; : means semidirect product.\n')

Arajiv1518 = rajiv1518.automorphism_group()
Aloz1404 = loz1404.automorphism_group()

print('rajiv1518 :', Arajiv1518.structure_description(), ' |  center order:',Arajiv1518.center().order())
print('loz1404   :', Aloz1404.structure_description(), ' |  center order:',Aloz1404.center().order())

print("\n")



# 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 nonisographs:
    print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}")

'''
# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 5")
for label, graph in nonisographs:
    print(f"{label} ","  &  ".join(str(count_k_cycles(graph, k)) for k in range(3, 6)))
'''



'''
# from 0 versio Molodtsov?

for name, G in nonisographs:
    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 nonisographs:
    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 nonisographs:
    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")        
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