# SageMathCell online  https://sagecell.sagemath.org/?q=qfqefk
#
#  Ord.: 16384 communicated Sep 18, 2026 /  Moore bound = 57857 
#  graph-lift construction found by J. Amarel, N. DeBardeleben, M. Grosskopf, V. Job, E. Lawrence, L. Monroe, R Somasundaram
#  by using  agentic tool URSA (https://lanl.github.io/ursa). LA-UR-26-28353

'''

  Main properties of the graph
deg16D4    | Ord.: 16384 / Size: 131072 / Diam.: 4 / Avg.dist: 3.82152 / 16-reg.? True /  Girth: 4  
 Symmetry properties of the graph
deg16D4    | Aut.group.ord.: 48 /  Cayley ? False ---  vtx.trans. ? False -- edge.trans. ? False
 Automorphism group structure
 C_n is the cyclic group of order n;  x means direct product; : means semidirect product.
deg16D4: C24 x C2  |  center order: 48
deg16D4    distance distrib from vtx. 0: [1, 16, 203, 2466, 13698]
Number of k-cycles for k=3 up to 4
deg16D4     0 | 299008
'''


import networkx as nx
deg16D4  = 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                
deg16D4nx = deg16D4.networkx_graph()

# List of graphs to process
graphs = [('deg16D4   ',deg16D4)]


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)} / 16-reg.? {graph.is_regular(k=16)} /  Girth: {graph.girth()}  ")#  /  Alg.conn. {algebraic_connectivity(graph).n(digits=6)} ")# /   Domin. number: {graph.dominating_set(value_only=True)} ")

#print("Degree histogram Q8pxX8p:", nx.degree_histogram(Q8pxX8pnx))  

    
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()} \\\\ ")
''' 
    
# 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')

Adeg16D4 = deg16D4.automorphism_group()
print('deg16D4:', Adeg16D4.structure_description(), ' |  center order:',Adeg16D4.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 4")
for label, graph in nonisographs:
    print(f"{label} "," | ".join(str(count_k_cycles(graph, k)) for k in range(3, 5)))

nx.write_edgelist(deg16D4nx, "deg16D4_edges.txt", data=False)

'''
# 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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