Some new bounds on the spectral radius and the energy of graphs
Resumo
Let $G=(V,E)$ be a simple connected graph. For any $v\in V$, the maximum neighbourhood degree $m(v)$ is the maximum vertex degree in the neighbourhood $N(v)$. In this paper, we present some new bounds on the spectral radius $\lambda_{1}(G)$ of $G$ in terms of degrees and maximum neighbourhood degrees of graph $G$. Also, we present some new bounds on the energy $E(G)$ of $G$.
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Referências
References
[1] Bar-Noy, A., Choudhary, K., Peleg, D. and Rawitz, D., Graph Realizations: Maximum and Minimum Degree in Vertex Neighborhoods, arXiv:
1912.13286v1(2019)
[2] Barrus, M., Donovan, E., Neighbourhood degree lists of graphs, Discret.Math. 341, 175-183(2018).
[3] Collatz, L. and Sinogowitz, U., Spektren Endlicher Grafen, Abh. Math.Sem. Univ. Hamburg 21, 63-77(1957).
[4] Cvetkovic, D., Doob, M. and Sachs, H., Spectra of Graphs: Theory and Applications, Academic Press, New York, 1980.
[5] Cvetkovic, D. and Rowlinson, P., The largest eigenvalue of a graph: A survey, Linear Multilinear Algebra 28, 3-33(1990).
[6] Das, K. Ch., Kumar, P., Some new bounds on the spectral radius of graphs, Discrete Mathematics 281, 149-161(2004).
[7] Edwards, C. and Elphick, C., Lower bounds for the clique and the chromatic number of a graph, Discr. Appl. Math. 5, 51-64(1983).
[8] Favaron, O., Maheo, M., J-FSacle, Some eigenvalue properties in graphs, Discrete. Math. 111, 179-220(1993).
[9] Randic, O. M., On Characterization of molecular branching, J. Amer. Chem. Soc. 97, 6609-6615(1975).
[10] Gutman, I., The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz. 103, 1-22(1978).
[11] Gutman, I., Polansky, O. E., Mathematical Concepts in Organic Chemistry, Springer, Berlin 1986.
13
[12] Gutman, I., Total electron energy of benzenoid hydrocarbons, Topics Curr. Chem 162, 29-63(1992).
[13] Gutman, I., The energy of a graph: Old and new results: A. Betten, A. Kohnert, R. Laue, A. Wassermann(Eds.), Springer Verlag,, 196-211(2001).
[14] Hefmeister, M., Spectral radius and degree sequence, Math. Nachr 139, 37-44(1988).
[15] Hou, Y., Tang, Z. andWoo, C., On the Spectral radius, k-degree and the upper bound of Energy in a Graph, MATCH Communn. Math. Comput.
Chem. 57, 341-350(2007).
[16] Li, X., Shi, Y., Gutman, I., Graph Energy, New York: Spinger (2012)
[17] MeClelland, B. J., Properties of the latent roots of a matrix: The estimation of -electron energies, J. Chem. Phys. 54, 640-643(1971).
[18] Nikiforov, V., Some inequalities for the largest eigenvalues of a graph, Combin. Probab. Comput. 11, 179-189(2003).
[19] Nikiforov, V., Walks and the spectral radius of graphs, Linear Algebra. Appl. 418, 257-268(2006).
[20] Nishimura, N. and Subramanya, V., Graph editing to given neighbourhood degree list is xed-parameter tractable, In X.Gao, H.Du
and M.Han(eds)COCOA 2017: Combinatorial Optimizations and applications, Vol.10628 of lecture Notes in Computer Science, 138-
153(2017)(Springer, Cham,).
[21] Stanley, R. P., A bound on the spectral radius with e edges, Linear Algebra. Appl. 87, 267-269(1987).
[22] Yu, A., Lu, M., Tian, F., On the spectral radius of graphs, Linear Algebra. Appl. 387, 41-49(2004).
[1] Bar-Noy, A., Choudhary, K., Peleg, D. and Rawitz, D., Graph Realizations: Maximum and Minimum Degree in Vertex Neighborhoods, arXiv:
1912.13286v1(2019)
[2] Barrus, M., Donovan, E., Neighbourhood degree lists of graphs, Discret.Math. 341, 175-183(2018).
[3] Collatz, L. and Sinogowitz, U., Spektren Endlicher Grafen, Abh. Math.Sem. Univ. Hamburg 21, 63-77(1957).
[4] Cvetkovic, D., Doob, M. and Sachs, H., Spectra of Graphs: Theory and Applications, Academic Press, New York, 1980.
[5] Cvetkovic, D. and Rowlinson, P., The largest eigenvalue of a graph: A survey, Linear Multilinear Algebra 28, 3-33(1990).
[6] Das, K. Ch., Kumar, P., Some new bounds on the spectral radius of graphs, Discrete Mathematics 281, 149-161(2004).
[7] Edwards, C. and Elphick, C., Lower bounds for the clique and the chromatic number of a graph, Discr. Appl. Math. 5, 51-64(1983).
[8] Favaron, O., Maheo, M., J-FSacle, Some eigenvalue properties in graphs, Discrete. Math. 111, 179-220(1993).
[9] Randic, O. M., On Characterization of molecular branching, J. Amer. Chem. Soc. 97, 6609-6615(1975).
[10] Gutman, I., The energy of a graph, Ber. Math. Stat. Sekt. Forschungsz. Graz. 103, 1-22(1978).
[11] Gutman, I., Polansky, O. E., Mathematical Concepts in Organic Chemistry, Springer, Berlin 1986.
13
[12] Gutman, I., Total electron energy of benzenoid hydrocarbons, Topics Curr. Chem 162, 29-63(1992).
[13] Gutman, I., The energy of a graph: Old and new results: A. Betten, A. Kohnert, R. Laue, A. Wassermann(Eds.), Springer Verlag,, 196-211(2001).
[14] Hefmeister, M., Spectral radius and degree sequence, Math. Nachr 139, 37-44(1988).
[15] Hou, Y., Tang, Z. andWoo, C., On the Spectral radius, k-degree and the upper bound of Energy in a Graph, MATCH Communn. Math. Comput.
Chem. 57, 341-350(2007).
[16] Li, X., Shi, Y., Gutman, I., Graph Energy, New York: Spinger (2012)
[17] MeClelland, B. J., Properties of the latent roots of a matrix: The estimation of -electron energies, J. Chem. Phys. 54, 640-643(1971).
[18] Nikiforov, V., Some inequalities for the largest eigenvalues of a graph, Combin. Probab. Comput. 11, 179-189(2003).
[19] Nikiforov, V., Walks and the spectral radius of graphs, Linear Algebra. Appl. 418, 257-268(2006).
[20] Nishimura, N. and Subramanya, V., Graph editing to given neighbourhood degree list is xed-parameter tractable, In X.Gao, H.Du
and M.Han(eds)COCOA 2017: Combinatorial Optimizations and applications, Vol.10628 of lecture Notes in Computer Science, 138-
153(2017)(Springer, Cham,).
[21] Stanley, R. P., A bound on the spectral radius with e edges, Linear Algebra. Appl. 87, 267-269(1987).
[22] Yu, A., Lu, M., Tian, F., On the spectral radius of graphs, Linear Algebra. Appl. 387, 41-49(2004).
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2025-01-24
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