# SageMathCell    online  --> https://sagecell.sagemath.org/?q=qejakn
#
# Obtained by Rishabh Rajiv (rishabh.rajiv@alumni.ubc.ca -communicated July 4, 2026) 
# Cayley graph for the semidirect product Z_11 (x45) Z_89 with generators 
#   [0,44 ]<>[0,45 ]:[1,43 ]<>[10,3 ]:[1,52 ]<>[10,74 ]:[1,80 ]<>[10,18 ]:[2,8 ]<>[9,57 ]:[4,14 ]<>[7,43 ]:[4,50 ]<>[7,1 ] 
#  From  https://doi.org/10.5281/zenodo.21180071
#  you can download the adjacency list, the verifier (standard-library Python), and a paper describing the construction of the graph.
#
#  | Ord.: 979 / Size: 6853 / Diam.: 3 / Avg.dist: 2.78528 / 14-reg.? True /  Girth: 5  /  Alg.conn. 8.26530 
#  | Aut.group.ord.: 979 /  Cayley ? True ---  vtx.trans. ? True -- edge.trans. ? False
#  distance distrib from vtx. 0: [1, 14, 182, 782]
#  Number of k-cycles for k=3 up to 5 :    [  0  |  0  |  34265]
#

import networkx as nx

rajiv979 = 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rajiv979nx = rajiv979.networkx_graph()

# List of graphs to process
graphs = [ ('rajiv979   ', rajiv979)]

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(graphs):
    canon = {}
    for label, G in graphs:
        s6 = G.canonical_label().sparse6_string()
        canon.setdefault(s6, []).append(label.strip())

    for group in canon.values():
        if len(group) > 1:
            print("Isomorphic :", ", ".join(group))
        else:
            print("Unique      :", group[0])

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")

# Check isomorphisms

compare_graphs_list(graphs)

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

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

#check_all_isomorphisms(nonisographs)



twographs = [    ]

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)} / 14-reg.? {graph.is_regular(k=14)} /  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("\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()}  & {graph.is_edge_transitive()} \\\\  ")
#    print(f"{label}   &   {graph.average_distance().n(digits=6)}  &   {graph.girth()}    &   {algebraic_connectivity(graph).n(digits=6)}   &    {domination_number_cayley(graph,solver= None)}   &    {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

print("\n")
for label, graph in nonisographs:
    print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}")

print("\n")
    

'''
print("\n")
print("\nChecking throught the spectrum that P216x4719 and P216x2202  are NON-isomorphic\n ")
for label, graph in twographs:
    spec = [ev.n(digits=4) for ev in graph.spectrum()[:20]]
    print(f"{label} spectrum (first 20): {spec}")
'''

# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 6")
for label, graph in nonisographs:
    print(f"{label} ", "  |  ".join(str(count_k_cycles(graph, k)) for k in range(3, 7)))



#nx.write_adjlist(MC216nx,"MC216_adjlist.txt")
#nx.write_adjlist(SG216x90nx,"SG216x90_adjlist.txt")
#nx.write_adjlist(SG216x158nx,SG216x15_adjlist.txt")

'''
# 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")

##