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4-tuple adjacency matrix algorithm Boneh-Boyen Cartesian product Cay(G cell centrality metrics Cm x Cn coefficients complete bipartite graph complete double vertex complete graph compute CONGRESSUS NUMERANTIUM 188 connected graph consider contradiction Corollary cubic graph cycle cover cyclic D(ia defined degree denote digraph disks double torus double vertex graph edge-magic exist integers Figure finite flipping follows forest in G friendly index friendly labeling G contains G is obtained G Zk GAMA glasses graceful labeling graph G graph of order Graph Theory Hall's condition Hamiltonian Hence hyperelliptic curve independent decycling set independent set induced jacobian jacobian variety Lemma Let G line-pairs longest path maximum mixing cycles neighborhood sum nodes path in G positive integer Proof Proposition satisfied Sheffer sequence simple graph spanning linear forest subgraph Suppose Tate pairing Tate's pairing Theorem travels triangulation triangulation G vertex set vertices of G