Brondani, Andre E.; França, Francisca Andrea Macedo; Oliveira, Carla S. \(A_\alpha\) and \(L_\alpha\)-spectral properties of spider graphs. (English) Zbl 1496.05095 Proyecciones 41, No. 4, 965-982 (2022). Summary: Let \(G\) be a graph with adjacency matrix \(A(G)\) and let \(D(G)\) be the diagonal matrix of the degrees of \(G\). For every real \(\alpha \in [0, 1]\), V. Nikiforov [Appl. Anal. Discrete Math. 11, No. 1, 81–107 (2017; Zbl 1499.05384)] and S. Wang et al. [Linear Algebra Appl. 590, 210–223 (2020; Zbl 1437.05151)] defined the matrices \(A_\alpha(G)\) and \(L_\alpha(G)\), respectively, as \(A_\alpha(G) = \alpha D(G)+(1- \alpha) A(G)\) and \(L_\alpha(G) = \alpha D(G)+(\alpha - 1) A(G)\). In this paper, we obtain some relationships between the eigenvalues of these matrices for some families of graphs, a part of the \(A_\alpha\) and \(L_\alpha\)-spectrum of the spider graphs, and we display the \(A_\alpha\) and \(L_\alpha\)-characteristic polynomials when their set of vertices can be partitioned into subsets that induce regular subgraphs. Moreover, we determine some subfamilies of spider graphs that are cospectral with respect to these matrices. MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A18 Eigenvalues, singular values, and eigenvectors Keywords:\(A_\alpha\)-spectrum; \(L_\alpha\)-spectrum; regular graphs; spider graphs Citations:Zbl 1437.05151; Zbl 1499.05384 PDFBibTeX XMLCite \textit{A. E. Brondani} et al., Proyecciones 41, No. 4, 965--982 (2022; Zbl 1496.05095) Full Text: DOI References: [1] N. M. M. Abreu, “Old and new results on algebraic connectivity of graphs”, Linear Algebra and its Applications, vol. 423, no. 1, pp. 53-73, 2007. · Zbl 1115.05056 [2] N. M. M. Abreu, “Old and new results on algebraic connectivity of graphs”, Linear Algebra and its Applications, vol. 423, no. 1, pp. 53-73, 2007. doi: 10.1016/j.laa.2006.08.017 · Zbl 1115.05056 [3] A. E. Brondani, C. S. Oliveira, F. A. M. França and L. de Lima, “Aα-Spectrum of a Firefly Graph”, Electronic Notes in Theoretical Computer Science, vol. 346, pp. 209-219, 2019. doi: 10.1016/j.entcs.2019.08.019 · Zbl 07515181 [4] A. E. Brondani, C. S. Oliveira, F. A. M. França and L. de Lima, “Aα-Spectrum of a Firefly Graph”, Electronic Notes in Theoretical Computer Science, vol. 346, pp. 209-219, 2019. · Zbl 07515181 [5] A. E. Brouwer and W. H. Haemers, Spectra of Graphs. New York: Springer, 2012. · Zbl 1231.05001 [6] A. E. Brouwer and W. H. Haemers, Spectra of Graphs. New York: Springer, 2012. · Zbl 1231.05001 [7] M. Cámara and W. H. Haemers, “Spectral characterizations of almost complete graphs”, Discrete Applied Mathematics, vol. 176, pp. 19-23, 2014. doi: 10.1016/j.dam.2013.08.002 · Zbl 1298.05233 [8] M. Cámara and W. H. Haemers, “Spectral characterizations of almost complete graphs”, Discrete Applied Mathematics, vol. 176, pp. 19-23, 2014. · Zbl 1298.05233 [9] D. G. Corneil, H. Lerchs and L. S. Burlingham, “Complement reducible graphs”, Discrete Applied Mathematics, vol. 3, no. 3, pp. 163-174, 1981. · Zbl 0463.05057 [10] D. G. Corneil, H. Lerchs and L. S. Burlingham, “Complement reducible graphs”, Discrete Applied Mathematics, vol. 3, no. 3, pp. 163-174, 1981. doi: 10.1016/0166-218x(81)90013-5 · Zbl 0463.05057 [11] D. M. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press, 2010. · Zbl 1211.05002 [12] D. M. Cvetković, P. Rowlinson and S. Simić, An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press, 2010. · Zbl 1211.05002 [13] D. M. Cvetković, “Graphs and their spectra”, Publikacije Elektrotehnickog Fakulteta Univerzitet Beograd, Serija Matematika, vol. 3, pp. 1-50, 1971. · Zbl 0238.05102 [14] D. M. Cvetković, “Graphs and their spectra”, Publikacije Elektrotehnickog Fakulteta Univerzitet Beograd, Serija Matematika, vol. 3, pp. 1-50, 1971. · Zbl 0238.05102 [15] K. Ch. Das and M. Liu, “Complete Split graph determined by its (singless) Laplacian spectrum”, Discrete Applied Mathematics, vol. 205, pp. 45-51, 2016. · Zbl 1333.05180 [16] K. Ch. Das and M. Liu, “Complete Split graph determined by its (singless) Laplacian spectrum”, Discrete Applied Mathematics, vol. 205, pp. 45-51, 2016. doi: 10.1016/j.dam.2016.01.003 · Zbl 1333.05180 [17] F. Goldberg, S. Kirkland, A. Varghese and A. Vijayakumar, “On split graphs with four distinct eigenvalues”, Discrete Applied Mathematics, vol. 277, pp. 163-171, 2020. doi: 10.1016/j.dam.2019.09.016 · Zbl 1435.05128 [18] F. Goldberg, S. Kirkland, A. Varghese and A. Vijayakumar, “On split graphs with four distinct eigenvalues”, Discrete Applied Mathematics, vol. 277, pp. 163-171, 2020. · Zbl 1435.05128 [19] H. J. Finck, “Vollstandiges Produkt, chromatische Zahl und characteristisches Polynom regulärer Graphen II”, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau, vol. 11, pp. 81-87, 1965. · Zbl 0132.20802 [20] H. J. Finck, “Vollstandiges Produkt, chromatische Zahl und characteristisches Polynom regulärer Graphen II”, Wissenschaftliche Zeitschrift der Technischen Hochschule Ilmenau, vol. 11, pp. 81-87, 1965. · Zbl 0132.20802 [21] M. Ghorbani and N. Azimi, “Characterization of split graphs with at most four distinct eigenvalues”, Discrete Applied Mathematics, vol. 184, pp. 231-236, 2015. · Zbl 1311.05112 [22] M. Ghorbani and N. Azimi, “Characterization of split graphs with at most four distinct eigenvalues”, Discrete Applied Mathematics, vol. 184, pp. 231-236, 2015. doi: 10.1016/j.dam.2014.10.039 · Zbl 1311.05112 [23] Hs. H. Günthard and H. Primas, “Zusammenhang von Graph theorie and MO-Theotie von Molekeln mit Systemen konjugierter Bindungen”, Helvetica Chimica Acta, vol. 39, no. 6, pp. 1645-1653, 1956. doi: 10.1002/hlca.19560390623 [24] Hs. H. Günthard and H. Primas, “Zusammenhang von Graph theorie and MO-Theotie von Molekeln mit Systemen konjugierter Bindungen”, Helvetica Chimica Acta, vol. 39, no. 6, pp. 1645-1653, 1956. [25] W. H. Haemers, “Cospectral pairs of regular graphs with different connectivity”, Discussiones Mathematicae Graph Theory, vol. 40, no. 2, pp. 577-584, 2020. doi: 10.7151/dmgt.2278 · Zbl 1433.05195 [26] W. H. Haemers, “Cospectral pairs of regular graphs with different connectivity”, Discussiones Mathematicae Graph Theory, vol. 40, no. 2, pp. 577-584, 2020. · Zbl 1433.05195 [27] M. Haythorpe, A. Newcombe, “Constructing families of cospectral regular graphs”, Combinatorics, Probability and Computing, vol. 29, no. 5, pp. 664-671, 2020. doi: 10.1017/s096354832000019x · Zbl 1462.05226 [28] M. Haythorpe, A. Newcombe, “Constructing families of cospectral regular graphs”, Combinatorics, Probability and Computing, vol. 29, no. 5, pp. 664-671, 2020. · Zbl 1462.05226 [29] C. T. Hoàng, Perfect graph, Thesis Ph.D. Canada: McGill University, Montreal, 1985. [30] C. T. Hoàng, Perfect graph, Thesis Ph.D. Canada: McGill University, Montreal, 1985. [31] B. Jamison and S. Olariu, “A tree representation for P4-sparse graphs”, Discrete Applied Mathematics, vol. 35, no. 2, pp. 115-129, 1992. · Zbl 0763.05092 [32] B. Jamison and S. Olariu, “A tree representation for P 4 -sparse graphs”, Discrete Applied Mathematics, vol. 35, no. 2, pp. 115-129, 1992. doi: 10.1016/0166-218x(92)90036-a · Zbl 0763.05092 [33] H. Q. Lin and J. L. Shu, “On the signless Laplacian index of cacti with a given number of pendant vertices”, Linear Algebra and its Applications, vol. 436, no. 12, pp. 4400-4411, 2012. · Zbl 1241.05082 [34] H. Q. Lin and J. L. Shu, “On the signless Laplacian index of cacti with a given number of pendant vertices”, Linear Algebra and its Applications, vol. 436, no. 12, pp. 4400-4411, 2012. doi: 10.1016/j.laa.2011.03.065 · Zbl 1241.05082 [35] L. Mao, S. M Cioaba and W. Wang, “Spectral characterizations of the complete graph removing a path of small length”, Discrete Applied Mathematics, vol. 257, pp. 260-268, 2019. doi: 10.1016/j.dam.2018.08.029 · Zbl 1406.05065 [36] L. Mao, S. M Cioaba and W. Wang, “Spectral characterizations of the complete graph removing a path of small length”, Discrete Applied Mathematics, vol. 257, pp. 260-268, 2019. · Zbl 1406.05065 [37] R. Medina, C. Noyer and O. Raynaud, “Twins Vertices in Hypergraphs”, Electronic Notes in Discrete Mathematics, vol. 27, pp. 87-89, 2006. · Zbl 1293.05248 [38] R. Medina, C. Noyer and O. Raynaud, “Twins Vertices in Hypergraphs”, Electronic Notes in Discrete Mathematics, vol. 27, pp. 87-89, 2006. doi: 10.1016/j.endm.2006.08.069 · Zbl 1293.05248 [39] H. Nagarajan, S. Rathinam and S. Darbha, “On maximizing algebraic connectivity of networks for various engineering applications”, European Control Conference (ECC), pp. 1626-1632, 2015. [40] H. Nagarajan, S. Rathinam and S. Darbha, “On maximizing algebraic connectivity of networks for various engineering applications”, European Control Conference (ECC), pp. 1626-1632, 2015. doi: 10.1109/ecc.2015. 7330770 [41] V. Nikiforov, “Merging the A- and Q-Spectral Theories”, Applicable Analysis and Discrete Mathematics, vol. 11, no. 1, pp. 81-107, 2017. · Zbl 1499.05384 [42] V. Nikiforov, “Merging the A-and Q-Spectral Theories”, Applicable Analysis and Discrete Mathematics, vol. 11, no. 1, pp. 81-107, 2017. doi: 10.2298/aadm1701081n · Zbl 1499.05384 [43] A. Seress, “Large Families of Cospectral Graphs”, Designs, Codes and Cryptography, vol. 21, pp. 205-208, 2000. · Zbl 0972.05033 [44] A. Seress, “Large Families of Cospectral Graphs”, Designs, Codes and Cryptography, vol. 21, pp. 205-208, 2000. doi: 10.1023/A:1008352030960 · Zbl 0972.05033 [45] W. So, “Commutativity and spectra of Hermitian matrices”, Linear Algebra and its Applications, vol. 212-213, pp. 121-129, 1994. · Zbl 0815.15005 [46] W. So, “Commutativity and spectra of Hermitian matrices”, Linear Algebra and its Applications, vol. 212-213, pp. 121-129, 1994. doi: 10.1016/0024-3795(94)90399-9 · Zbl 0815.15005 [47] A. Sorokin, R. Murphey, My T. Thai and P. M. Pardalos, Dynamics of Information Systems Mathematical Foundations. New York: Springer, 2012. · Zbl 1255.93007 [48] A. Sorokin, R. Murphey, My T. Thai and P. M. Pardalos, Dynamics of Information Systems Mathematical Foundations. New York: Springer, 2012. · Zbl 1255.93007 [49] E. R. van Dam and W. H. Haemers, “Which graphs are determined by their spectrum?”, Linear Algebra and its Applications, vol. 373, no. 1, pp. 241-272, 2003. doi: 10.1016/s0024-3795(03)00483-x · Zbl 1026.05079 [50] E. R. van Dam and W. H. Haemers, “Which graphs are determined by their spectrum?”, Linear Algebra and its Applications, vol. 373, no. 1, pp. 241-272, 2003. · Zbl 1026.05079 [51] S. Wang and D. Wong, F. Tian, “Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees”, Linear Algebra and its Applications, vol. 590, pp. 210-223, 2020. · Zbl 1437.05151 [52] S. Wang and D. Wong, F. Tian, “Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees”, Linear Algebra and its Applications, vol. 590, pp. 210-223, 2020. doi: 10.1016/j.laa.2019.12.039 · Zbl 1437.05151 [53] P. Wei and D. Sun, “Weighted Algebraic Connectivity: An Application to Airport Transportation Network”, Proceedings of the 18th World Congress The International Federation of Automatic Control, pp. 13864-13869, 2011. doi: 10.3182/20110828-6-it-1002.00486 [54] P. Wei and D. Sun, “Weighted Algebraic Connectivity: An Application to Airport Transportation Network”, Proceedings of the 18th World Congress The International Federation of Automatic Control, pp. 13864-13869, 2011. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.