Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G)>=5 has a Hamiltonian circuit if and only if the matrix A+I can be permuted on rows such that each column has at most (or exactly) k-1 circular blocks of consecutive ones; and if the graph G is k-regular except for two (k-1)-degree vertices a and b, then there is a Hamiltonian path from a to b if and only if the matrix A+I can be permuted on rows to have at most (or exactly) k-1 linear blocks per column. Then  turn to the problem of determining whether a given matrix can have at most k blocks of consecutive ones per column by some row permutation. For this problem, Booth and Lueker gave a linear algorithm for k=1 [Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, 1975, pp. 255-265]; Flammini et al. shod its NP-completeness for general k [Algorithmica 16 (1996) 549-568]; and Goldberg et al. proved the same for every fixed k>=2 [J. Comput. Biol. 2 (1) (1995) 139-152]. ,  strengthen their result by proving that the problem remains NP-complete for every constant k>=2 even if the matrix is restricted to (1) symmetric, or (2) having at most three blocks per row.
['Rui Wang', 'Francis C. M. Lau', 'Yingchao Zhao']
I am writing a research paper on a idea Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices
[{"from": "human", "value": "I am writing a research paper on a idea Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G)>=5 has a Hamiltonian circuit if and only if the matrix A+I can be permuted on rows such that each column has at most (or exactly) k-1 circular blocks of consecutive ones; and if the graph G is k-regular except for two (k-1)-degree vertices a and b, then there is a Hamiltonian path from a to b if and only if the matrix A+I can be permuted on rows to have at most (or exactly) k-1 linear blocks per column. Then  turn to the problem of determining whether a given matrix can have at most k blocks of consecutive ones per column by some row permutation. For this problem, Booth and Lueker gave a linear algorithm for k=1 [Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, 1975, pp. 255-265]; Flammini et al. shod its NP-completeness for general k [Algorithmica 16 (1996) 549-568]; and Goldberg et al. proved the same for every fixed k>=2 [J. Comput. Biol. 2 (1) (1995) 139-152]. ,  strengthen their result by proving that the problem remains NP-complete for every constant k>=2 even if the matrix is restricted to (1) symmetric, or (2) having at most three blocks per row."}]