# SageMathCell online https://sagecell.sagemath.org/?q=jquzbq # Degree= 8, Diameter = 5; Order =5115; Moore bound=22409 (F. Comellas 2024) # # Cayley graph with 5115 vertices and 20460 edges # Semidirect product Z_15 x(267) Z_341 # generators [13,277]<>[2,257]:[1,290]<>[14,50]:[4.21]<>[11,69]:[10,258]<>[5,105] # Degree Distribution: [0, 0, 0, 0, 0, 0, 0, 0, 5115] # avg. deg.: 8.0 max. deg.: 8 # Diameter: 5 avg. dist.: 4.378178 (1,8,56,392,2196,2462) transm.: 22390 Girth: 7 # # import networkx as nx Com5115 = 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Com5115nx =Com5115.networkx_graph() # List of graphs to process graphs = [('Com5115 ', Com5115)] 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(Com5115nx)} / 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()}" ) # 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)}") # Counting k-cycles for each graph print("\nNumber of k-cycles for k=3 up to 7") for label, graph in graphs: print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 8))) print("\n") ##