×

Graphs with least eigenvalue \(-2\): ten years on. (English) Zbl 1326.05079

Summary: The authors’ monograph [Spectral generalizations of line graphs. On graphs with least eigenvalue \(-2\). Cambridge: Cambridge University Press (2004; Zbl 1061.05057)] was published in 2004, following the successful use of star complements to complete the classification of graphs with least eigenvalue \(-2\). Guided by citations of the book, we survey progress in this area over the past decade. Some new observations are included.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)

Citations:

Zbl 1061.05057

Software:

ILIGRA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balińska, K. T.; Simić, S. K.; Zwierzyński, K. T., Which nonregular, bipartite, integral graphs with maximum degree four do not have ±1 as eigenvalues?, Discrete Math., 286, 15-25 (2004) · Zbl 1048.05056
[2] Bell, F. K.; Li Marzi, E. M.; Simić, S. K., Some new results on graphs with least eigenvalue not less than −2, Rend. Sem. Mat. Messina Ser. II, 9, 11-30 (2003) · Zbl 1124.05314
[3] Branković, Lj.; Cvetković, D., The eigenspace of the eigenvalue −2 in generalized line graphs and a problem in security of statistical data bases, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 14, 37-48 (2003) · Zbl 1105.05041
[4] Cvetković, D.; Rowlinson, P.; Simić, S., Graphs with least eigenvalue −2; a new proof of the 31 forbidden subgraphs theorem, Des. Codes Cryptogr., 34, 229-240 (2005) · Zbl 1063.05090
[5] Rowlinson, P., Star complements and the maximal exceptional graphs, Publ. Inst. Math. (Beograd) (N.S.), 76, 90, 25-30 (2004) · Zbl 1088.05053
[6] Stevanović, D., 4-regular integral graphs avoiding ±3 in the spectrum, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 14, 99-110 (2003) · Zbl 1088.05504
[7] Brouwer, A. E.; Haemers, W. H., Spectra of Graphs (2011), Springer: Springer New York · Zbl 0794.05076
[8] Cvetković, D.; Doob, M.; Sachs, H., Spectra of Graphs; Theory and Application (1995), Johann Ambrosius Barth Verlag: Johann Ambrosius Barth Verlag Heidelberg-Leipzig · Zbl 0824.05046
[9] Cvetković, D.; Rowlinson, P.; Simić, S., Eigenspaces of Graphs (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0878.05057
[10] Cvetković, D.; Rowlinson, P.; Simić, S., Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue −2 (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1061.05057
[11] Cvetković, D.; Rowlinson, P.; Simić, S., An Introduction to the Theory of Graph Spectra (2009), Cambridge University Press: Cambridge University Press Cambridge
[12] Jost, J., Riemannian Geometry and Geometric Analysis (2011), Springer: Springer Heidelberg · Zbl 1227.53001
[13] Lehrer, G. I.; Taylor, D. E., Unitary Reflection Groups (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1189.20001
[14] van Mieghem, P., Graph Spectra for Complex Networks (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1232.05128
[15] Stekolshchik, R., Notes on Coxeter Transformations and the McKay Correspondence (2008), Springer: Springer Berlin · Zbl 1202.20045
[16] Stanić, Z., Some reconstructions in spectral graph theory and graphs with integral Q-spectrum (2007), Faculty of Mathematics, University of Belgrade, (in Serbian), PhD thesis
[17] Gumbrell, L., On spectral constructions for Salem graphs (2013), Royal Holloway, University of London, PhD Thesis
[18] Koledin, T., Some classes of spectrally constrained graphs (2013), Faculty of Mathematics, University of Belgrade, (in Serbian), PhD thesis
[19] de Abreu, N. M.M., Old and new results on algebraic connectivity of graphs, Linear Algebra Appl., 423, 53-73 (2007) · Zbl 1115.05056
[20] de Abreu, N. M.M.; Balińska, K. T.; Simić, S. K., More on non-regular bipartite graphs with maximum degree four not having ±1 as eigenvalues, Appl. Anal. Discrete Math., 8, 123-154 (2014) · Zbl 1349.05227
[21] Akbari, S.; Ghorbani, E.; Mahmoodi, A., On graphs whose star sets are (co-)cliques, Linear Algebra Appl., 430, 504-510 (2009) · Zbl 1168.05324
[22] Andelić, M.; Cardoso, D. M.; Simić, S. K., Relations between \((\kappa, \tau)\)-regular sets and star complements, Czechoslovak Math. J., 63, 138, 73-90 (2013) · Zbl 1274.05286
[23] Arsić, B.; Cvetković, D.; Simić, S. K.; Škarić, M., Graph spectral techniques in computer sciences, Appl. Anal. Discrete Math., 6, 1-30 (2012) · Zbl 1289.05266
[24] Bapat, R. B., A note on singular line graphs, Bull. Kerala Math. Assoc., 8, 207-209 (2011)
[25] Barbedo, I.; Cardoso, D. M.; Cvetković, D.; Rama, P.; Simić, S. K., A recursive construction of regular exceptional graphs with least eigenvalue −2, Port. Math., 71, 79-96 (2014) · Zbl 1297.05142
[26] Bell, F. K.; Cvetković, D.; Rowlinson, P.; Simić, S. K., Graphs for which the least eigenvalue is minimal I, Linear Algebra Appl., 429, 234-241 (2008) · Zbl 1149.05030
[27] Biggs, N. L., Some properties of strongly regular graphs (June 2011)
[28] Brankov, V.; Cvetković, D.; Simić, S. K.; Stevanović, D., Simultaneous editing and multilabelling of graphs in system newGRAPH, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 17, 112-121 (2006) · Zbl 1199.68191
[29] Brouwer, A. E.; Haemers, W. H., Hamiltonian strongly regular graphs · Zbl 0764.05098
[30] Cardoso, D. M.; Cvetković, D., Graphs with least eigenvalue −2 attaining a convex quadratic upper bound for the stability number, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 133, 42-55 (2006) · Zbl 1265.05353
[31] Cardoso, D. M.; Cvetković, D.; Rowlinson, P.; Simić, S. K., A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph, Linear Algebra Appl., 429, 2770-2780 (2008) · Zbl 1148.05046
[32] Cardoso, D. M.; Kaminski, M.; Lozin, V., Maximum \(k\)-regular induced subgraphs, J. Comb. Optim., 14, 4, 455-463 (2007) · Zbl 1149.90169
[33] Chung, T.; Koolen, J.; Sano, Y.; Taniguchi, T., The non-bipartite integral graphs with spectral radius three, Linear Algebra Appl., 435, 2544-2559 (2011) · Zbl 1222.05151
[34] Chang, T.-C.; Tam, B.-S.; Wu, S.-H., Theorems on partitioned matrices revisited and their applications to graph spectra, Linear Algebra Appl., 434, 559-581 (2011) · Zbl 1225.05160
[35] Cvetković, D., Graphs with least eigenvalue −2: the eigenspace of the eigenvalue −2, Rend. Sem. Mat. Messina Ser. II, 25, 9, 63-86 (2003) · Zbl 1124.05061
[36] Cvetković, D., Notes on maximal exceptional graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 15, 103-107 (2004)
[37] Cvetković, D., Signless Laplacians and line graphs, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 131, 85-92 (2005) · Zbl 1119.05066
[38] Cvetković, D.; Grout, J., Maximal energy graphs should have a small number of distinct eigenvalues, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 134, 43-57 (2007) · Zbl 1199.05213
[39] Cvetković, D.; Lepović, M., Sets of cospectral graphs with least eigenvalue at least −2 and some related results, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math., 129, 85-102 (2004) · Zbl 1077.05059
[40] Cvetković, D.; Lepović, M., Cospectral graphs with least eigenvalue at least −2, Publ. Inst. Math. (Beograd) (N.S.), 78, 92, 51-63 (2005) · Zbl 1265.05358
[41] Cvetković, D.; Lepović, M., Towards an algebra of SINGs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 16, 110-118 (2005) · Zbl 1104.05043
[42] Cvetković, D.; Lepović, M., A table of cospectral graphs with least eigenvalue at least −2, (Cvetković, D., Iracionalno u racionalnom (2011), Akademska misao: Akademska misao Beograd), 173-196, see also
[43] Cvetković, D.; Rowlinson, P.; Simić, S. K., Star complements and exceptional graphs, Linear Algebra Appl., 423, 146-154 (2007) · Zbl 1124.05060
[44] Cvetković, D.; Rowlinson, P.; Simić, S. K., Signless Laplacians of finite graphs, Linear Algebra Appl., 423, 155-171 (2007) · Zbl 1113.05061
[45] Cvetković, D.; Rowlinson, P.; Simić, S. K., Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd) (N.S.), 81, 95, 11-27 (2007) · Zbl 1164.05038
[46] Cvetković, D.; Rowlinson, P.; Stanić, Z.; Yoon, M.-G., Controllable graphs with least eigenvalue at least −2, Appl. Anal. Discrete Math., 5, 165-175 (2011) · Zbl 1265.05359
[47] Cvetković, D.; Simić, S. K., Errata, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 15, 112 (2004)
[48] Cvetković, D.; Simić, S. K., Towards a spectral theory of graphs based on the signless Laplacian I, Publ. Inst. Math. (Beograd) (N.S.), 85, 99, 19-33 (2009) · Zbl 1224.05293
[49] Cvetković, D.; Simić, S. K., Towards a spectral theory of graphs based on the signless Laplacian II, Linear Algebra Appl., 432, 2257-2272 (2010) · Zbl 1218.05089
[50] Cvetković, D.; Simić, S. K., Graph spectra in computer science, Linear Algebra Appl., 434, 1545-1562 (2011) · Zbl 1207.68230
[51] Cvetković, D.; Simić, S. K., Spectral graph theory in computer science, IPSI BgD Trans. Adv. Res., 8, 35-42 (2012)
[52] Erić, A.; da Fonseca, C. M., Some consequences of an inequality on the spectral multiplicity of graphs, Filomat, 27, 1455-1461 (2013) · Zbl 1392.05074
[53] Fan, Y.-Z.; Zhang, F.-F.; Wang, Y., The least eigenvalue of the complements of trees, Linear Algebra Appl., 435, 2150-2155 (2011) · Zbl 1222.05156
[54] Fiol, M. A.; Mitjana, M., The local spectra of line graphs, (6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications. 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Electron. Notes Discrete Math., vol. 28 (2007), Elsevier Sci. B.V.: Elsevier Sci. B.V. Amsterdam), 95-102 · Zbl 1291.05121
[55] Fiol, M. A.; Mitjana, M., The local spectra of regular line graphs, Discrete Math., 310, 511-517 (2010) · Zbl 1215.05103
[56] Fricke, G. K.; Rogers, B. W.; Garg, D. P., On the stability of swarm consensus under noisy control, (Proc. ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control, vol. 1. Proc. ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control, vol. 1, Arlington, Virginia, USA (2011)), 291-298
[57] Gernert, D.; Rabern, L., A computerized system for graph theory, illustrated by partial proofs for graph-coloring problems, Graph Theory Notes N. Y., 22, 14-24 (2008)
[58] Ghorbani, E., Spanning trees and even integer eigenvalues of graphs, Discrete Math., 324, 62-67 (2014) · Zbl 1284.05160
[59] Greaves, G.; Koolen, J.; Munemasa, A.; Sano, Y.; Taniguchi, T., Edge-signed graphs with smallest eigenvalue greater than −2, J. Combin. Theory Ser. B, 110, 90-111 (2015) · Zbl 1302.05074
[60] Gumbrell, L., A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues (2012)
[61] Gumbrell, L.; McKee, J., A classification of all 1-Salem graphs, LMS J. Comput. Math., 17, 582-594 (2014) · Zbl 1346.05164
[62] Gutman, I.; Robbiano, M.; Martins, E. A.; Cardoso, D. M.; Medina, L.; Rojo, O., Energy of line graphs, Linear Algebra Appl., 433, 1312-1323 (2010) · Zbl 1194.05137
[63] Haemers, W. H., Matrices and graphs (2005), Tilburg University, or · Zbl 0745.51003
[64] Haemers, W.; Spence, E., Enumeration of cospectral graphs, European J. Combin., 25, 199-211 (2004) · Zbl 1033.05070
[65] Huang, L.-H.; Tam, B.-S.; Wu, S.-H., Graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows, Linear Algebra Appl., 438, 4008-4040 (2013) · Zbl 1282.05120
[66] Jaklič, G.; Fowler, P. W.; Pisanski, T., The HL-index of a graph, Ars Math. Contemp., 5, 99-105 (2012) · Zbl 1244.05144
[67] Jang, H. J.; Koolen, J.; Munemasa, A.; Taniguchi, T., On fat Hoffman graphs with smallest eigenvalue at least −3, Ars Math. Contemp., 7, 105-121 (2014) · Zbl 1301.05219
[68] Koledin, T.; Stanić, Z., Regular graphs whose second largest eigenvalue is at most 1, Novi Sad J. Math., 43, 3, 145-153 (2013) · Zbl 1299.05223
[69] Koolen, J. H.; Yu, H., The distance-regular graphs such that all of its second largest local eigenvalues are at most one, Linear Algebra Appl., 435, 2507-2519 (2011) · Zbl 1222.05163
[70] Krishnasamy, M.; Taylor, D. E., Embeddings of complex line systems and finite reflection groups, J. Aust. Math. Soc., 85, 211-228 (2008) · Zbl 1173.51003
[71] Lang, W.; Wang, L., Energy of generalized line graphs, Linear Algebra Appl., 437, 9, 2386-2396 (2012) · Zbl 1247.05138
[72] Lepović, M.; Simić, S. K.; Balińska, K. T.; Zwierzyński, K. T., There are 93 non-regular, bipartite integral graphs with maximum degree four (2005), The Technical University of Poznań: The Technical University of Poznań Poznań, CSC Report No. 511
[73] Li, H.-H.; Fan, Y.-Z.; Su, L., On the nullity of the line graph of unicyclic graph with depth one, Linear Algebra Appl., 437, 2038-2055 (2012) · Zbl 1247.05200
[74] Li, S.; Zhang, L., Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs, Linear Multilinear Algebra, 59, 145-158 (2011) · Zbl 1239.05116
[75] Li, S.; Zhang, L., Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter \(d\), Graphs Combin., 28, 531-546 (2012) · Zbl 1256.05141
[76] Liu, D.; Trajanovski, S.; Van Mieghem, P., Random line graphs and a linear law for assortativity, Phys. Rev. E, 87, 012816 (31 January 2013)
[77] Liu, D.; Trajanovski, S.; Van Mieghem, P., ILIGRA: an efficient inverse line graph algorithm, J. Math. Model. Algorithms, 14, 13-33 (2015) · Zbl 1347.05239
[78] Liu, Z., Energy, Laplacian energy and Zagreb index of line graph, middle graph and total graph, Int. J. Contemp. Math. Sci., 5, 895-900 (2010) · Zbl 1205.05148
[79] Marino, M. C.; Sciriha, I.; Simić, S. K.; Tošić, D. V., More about singular line graphs of trees, Publ. Inst. Math. (Beograd) (N.S.), 79, 93, 1-12 (2006) · Zbl 1121.05073
[80] McKee, J.; Smyth, C., Salem numbers, Pisot numbers, Mahler measure and graphs, Exp. Math., 14, 211-229 (2005) · Zbl 1082.11066
[81] Merajuddin, S. A.K.; Ali, P.; Pirzada, S., Cospectral and hyper-energetic self complementary comparability graphs, J. Korean Soc. Ind. Appl. Math., 11, 65-75 (2007)
[82] Milošević, M., An example of using star complements in classifying strongly regular graphs, Filomat, 22, 53-57 (2008) · Zbl 1199.05234
[83] Mohar, B.; Tayfeh-Rezaie, B., Median eigenvalues of bipartite graphs, J. Algebraic Combin., 41, 899-909 (2015) · Zbl 1317.05112
[84] Munemasa, A.; Sano, Y.; Taniguchi, T., Fat Hoffman graphs with smallest eigenvalue at least \(- 1 - \tau \), Ars Math. Contemp., 7, 247-262 (2014) · Zbl 1301.05221
[85] Munemasa, A.; Sano, Y.; Taniguchi, T., Fat Hoffman graphs with smallest eigenvalue greater than −3, Discrete Appl. Math., 176, 78-88 (2014) · Zbl 1298.05205
[86] Munemasa, A.; Sano, Y.; Taniguchi, T., On the smallest eigenvalues of the line graphs of some trees, Linear Algebra Appl., 466, 501-511 (2015) · Zbl 1302.05109
[87] Petrović, M.; Aleksić, T.; Simić, S. K., Further results on the least eigenvalue of connected graphs, Linear Algebra Appl., 435, 2303-2313 (2011) · Zbl 1222.05174
[88] Rašajski, M.; Radosavljević, Z.; Mihailović, B., Maximal reflexive cacti with four cycles: the approach via Smith graphs, Linear Algebra Appl., 435, 2530-2543 (2011) · Zbl 1222.05176
[89] Radosavljević, Z., On unicyclic reflexive graphs, Appl. Anal. Discrete Math., 1, 228-240 (2007) · Zbl 1199.05238
[90] Ramane, H. S.; Walikar, H. B.; Rao, S. B.; Acharya, B. D.; Hampiholi, P. R.; Jog, S. R.; Gutman, I., Spectra and energies of iterated line graphs of regular graphs, Appl. Math. Lett., 18, 679-682 (2005) · Zbl 1071.05551
[91] Rojo, O., Line graph eigenvalues and line energy of caterpillars, Linear Algebra Appl., 435, 2077-2086 (2011) · Zbl 1222.05177
[92] Rojo, O.; Jiménez, R. D., Line graph of combinations of generalized Bethe trees: eigenvalues and energy, Linear Algebra Appl., 435, 2402-2419 (2011) · Zbl 1222.05178
[93] Rowlinson, P., Co-cliques and star complements in extremal strongly regular graphs, Linear Algebra Appl., 421, 157-162 (2007) · Zbl 1116.05052
[94] Rowlinson, P., On multiple eigenvalues of trees, Linear Algebra Appl., 432, 3007-3011 (2010) · Zbl 1195.05044
[95] Rowlinson, P., On eigenvalue multiplicity and the girth of a graph, Linear Algebra Appl., 435, 2375-2381 (2011) · Zbl 1222.05179
[96] Rowlinson, P., Regular star complements in strongly regular graphs, Linear Algebra Appl., 436, 1482-1488 (2012) · Zbl 1236.05211
[97] Rowlinson, P., On induced matchings as star complements in regular graphs, J. Math. Sci., 182, 159-163 (2012) · Zbl 1254.05160
[98] Rowlinson, P., On graphs with an eigenvalue of maximal multiplicity, Discrete Math., 313, 1162-1166 (2013) · Zbl 1277.05109
[99] Rowlinson, P., Star complements and connectivity in finite graphs, Linear Algebra Appl., 442, 92-98 (2014) · Zbl 1282.05151
[100] Rowlinson, P., On independent star sets in finite graphs, Linear Algebra Appl., 442, 82-91 (2014) · Zbl 1282.05150
[101] Rowlinson, P., Eigenvalue multiplicity in cubic graphs, Linear Algebra Appl., 444, 211-218 (2014) · Zbl 1292.05179
[102] Rowlinson, P., Bipartite graphs with complete bipartite star complements, Linear Algebra Appl., 458, 149-160 (2014) · Zbl 1296.05125
[103] Rowlinson, P.; Sciriha, I., Some properties of the Hoffman-Singleton graph, Appl. Anal. Discrete Math., 1, 438-445 (2007) · Zbl 1199.05242
[104] Rowlinson, P.; Tayfeh-Rezaie, B., Star complements in regular graphs: old and new results, Linear Algebra Appl., 432, 2230-2242 (2010) · Zbl 1217.05156
[105] Sciriha, I., Repeated eigenvalues of the line graph of a tree and of its deck, Util. Math., 71, 33-55 (2006) · Zbl 1111.05064
[106] Sciriha, I.; Simić, S. K.; Buratti, M.; Lindner, C.; Mazzocca, F.; Melone, N., On eigenspaces of some compound graphs, A Tribute to Lucia Gionfriddo. A Tribute to Lucia Gionfriddo, Quad. Mat., 28, 123-154 (2013)
[107] Simić, S. K.; Andjelić, M.; da Fonseca, C. M.; Živković, D., On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results, Electron. J. Linear Algebra, 30 (2015), Article 6 · Zbl 1323.05083
[108] Simić, S. K.; Stanić, Z., The polynomial reconstruction is unique for the graphs whose deck-spectra are bounded from below by −2, Linear Algebra Appl., 428, 1865-1873 (2008) · Zbl 1135.05043
[109] Simić, S. K.; Stanić, Z., \(Q\)-integral graphs with edge-degrees at most five, Discrete Math., 308, 4625-4634 (2008) · Zbl 1156.05037
[110] Simić, S. K.; Stanić, Z., On \(Q\)-integral \((3, s)\)-semiregular bipartite graphs, Appl. Anal. Discrete Math., 4, 167-174 (2010) · Zbl 1289.05315
[112] Stanić, Z., On graphs whose second largest eigenvalue equals 1 - the star complement technique, Linear Algebra Appl., 420, 700-710 (2007) · Zbl 1106.05067
[113] Stanić, Z., Some star complements for the second largest eigenvalue of a graph, Ars Math. Contemp., 1, 126-136 (2008) · Zbl 1168.05332
[114] Stanić, Z., On nested split graphs whose second largest eigenvalue is less than 1, Linear Algebra Appl., 430, 2200-2211 (2009) · Zbl 1194.05098
[115] Stanić, Z., Some results on \(Q\)-integral graphs, Ars Combin., 90, 321-335 (2009) · Zbl 1224.05320
[116] Stanić, Z., On regular graphs and coronas whose second largest eigenvalue does not exceed 1, Linear Multilinear Algebra, 58, 545-554 (2010) · Zbl 1207.05124
[117] Stanić, Z., Graphs with small spectral gap, Electron. J. Linear Algebra, 26, 417-432 (2013) · Zbl 1282.05153
[119] Stevanović, D., Bipartite density of cubic graphs: the case of equality, Discrete Math., 283, 279-281 (2004) · Zbl 1042.05062
[120] Stevanović, D.; Milošević, M., A spectral proof of the uniqueness of a strongly regular graph with parameters (81, 20, 1, 6), European J. Combin., 30, 957-968 (2009) · Zbl 1207.05225
[121] Tan, Y.-Y.; Fan, Y.-Z., The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph, Linear Algebra Appl., 433, 790-795 (2010) · Zbl 1214.05085
[122] Taniguchi, T., On graphs with the smallest eigenvalue at least \(- 1 - \sqrt{2} \), part I, Ars Math. Contemp., 1, 81-98 (2008) · Zbl 1168.05333
[123] Taniguchi, T., On graphs with the smallest eigenvalue at least \(- 1 - \sqrt{2} \), part II, Ars Math. Contemp., 5, 239-254 (2012)
[124] Vijayakumar, G. R., Equivalence of four descriptions of generalized line graphs, J. Graph Theory, 67, 27-33 (2011) · Zbl 1226.05185
[125] Vijayakumar, G. R., Characterization of the family of all generalized line graphs, Discuss. Math. Graph Theory, 33, 637-648 (2013) · Zbl 1295.05197
[126] Vijayakumar, G. R., From finite line graphs to infinite derived signed graphs, Linear Algebra Appl., 453, 84-98 (2014) · Zbl 1314.05174
[127] Wang, J.-F.; Huang, Q., Spectral characterization of generalized cocktail-party graphs, J. Math. Res. Appl., 32, 666-672 (2012) · Zbl 1274.05306
[128] Wang, J.-F.; Shen, Y.; Huang, Q., Notes on graphs with least eigenvalue at least −2, Electron. J. Linear Algebra, 23, 387-396 (2012) · Zbl 1252.05135
[129] Wang, J.-F.; Shi, S., The line graphs of lollipop graphs are determined by their spectra, Linear Algebra Appl., 436, 2630-2637 (2012) · Zbl 1238.05167
[130] Wang, J.-F.; Yan, J., Comments to “The line graphs of lollipop graphs are determined by their spectra”, Linear Algebra Appl., 440, 342-344 (2014) · Zbl 1408.05084
[131] Wang, Y.; Fan, Y.-Z., The least eigenvalue of a graph with cut vertices, Linear Algebra Appl., 433, 19-27 (2010) · Zbl 1189.05107
[132] Wang, Y.; Fan, Y.-Z., The least eigenvalue of graphs with cut edges, Graphs Combin., 28, 555-561 (2012) · Zbl 1256.05144
[133] Wang, Y.; Fan, Y.-Z.; Li, X. X.; Zhang, F. F., The least eigenvalue of graphs whose complements are unicyclic, Discuss. Math. Graph Theory, 35, 249-260 (2015) · Zbl 1311.05119
[134] Yu, G.; Fan, Y., The least eigenvalue of graphs whose complements are 2-vertex or 2-edge connected, Oper. Res. Trans., 17, 81-88 (2013) · Zbl 1299.05238
[135] Yu, H., On the limit points of the smallest eigenvalues of regular graphs, Des. Codes Cryptogr., 65, 77-88 (2012) · Zbl 1245.05090
[136] Yuan, M. Y.; Luo, Q. H.; Tang, Z. K., Characterizing generalized line graphs by star complements (in Chinese), J. Nat. Sci. Hunan Norm. Univ., 35, 13-16 (2012), 20 · Zbl 1265.05524
[137] Zhang, H. R.; Wang, L. G., Constructing integral spectra graphs by super generalized line graph methods, Oper. Res. Trans., 15, 122-128 (2011), (in Chinese) · Zbl 1240.05259
[138] Zhou, B.; Ilić, A., On distance spectral radius and distance energy of graphs, MATCH Commun. Math. Comput. Chem., 64, 261-280 (2010) · Zbl 1265.05437
[139] Zhou, J.; Bu, C., Spectral characterization of line graphs of starlike trees, Linear Multilinear Algebra, 61, 1041-1050 (2013) · Zbl 1272.05169
[140] Zhou, J.; Sun, L.; Yao, H.; Bu, C., On the nullity of connected graphs with least eigenvalue at least −2, Appl. Anal. Discrete Math., 7, 250-261 (2013) · Zbl 1313.05260
[141] Bell, F. K.; Rowlinson, P., On the multiplicities of graph eigenvalues, Bull. Lond. Math. Soc., 35, 401-408 (2003) · Zbl 1023.05097
[142] Bollobás, B., Combinatorics (1986), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0595.05001
[143] Boyd, D. W., Small Salem numbers, Duke Math. J., 44, 315-328 (1977) · Zbl 0353.12003
[144] Cameron, P. J.; Goethals, J.-M.; Seidel, J. J.; Shult, E. E., Line graphs, root systems and elliptic geometry, J. Algebra, 43, 305-327 (1976) · Zbl 0337.05142
[145] Cameron, P. J.; van Lint, J. H., Designs, Graphs, Codes and Their Links (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0743.05004
[146] Chawathe, P. D.; Vijayakumar, G. R., A characterization of signed graphs represented by the root system \(D_\infty \), European J. Combin., 11, 523-533 (1990) · Zbl 0764.05090
[147] Cvetković, D.; Doob, M.; Simić, S., Generalized line graphs, J. Graph Theory, 5, 385-399 (1981) · Zbl 0475.05061
[148] Cvetković, D.; Petrić, M., A table of connected graphs on six vertices, Discrete Math., 50, 37-49 (1984) · Zbl 0533.05052
[149] Cvetković, D.; Stevanović, D., Graphs with least eigenvalue at least \(- \sqrt{3} \), Publ. Inst. Math. (Beograd) (N.S.), 73, 87, 39-51 (2003) · Zbl 1265.05361
[150] van Dam, E. R.; Haemers, W., Which graphs are determined by their spectrum?, Linear Algebra Appl., 373, 241-272 (2003) · Zbl 1026.05079
[151] Doob, M.; Cvetković, D., On spectral characterizations and embeddings of graphs, Linear Algebra Appl., 27, 17-26 (1979) · Zbl 0417.05025
[152] Grone, R.; Merris, R.; Sunder, V. S., The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl., 11, 218-238 (1990) · Zbl 0733.05060
[153] Gutman, I.; Sciriha, I., On the nullity of line graphs of trees, Discrete Math., 232, 35-45 (2001) · Zbl 0971.05070
[154] Haemers, W., Interlacing eigenvalues and graphs, Linear Algebra Appl., 226-228, 593-616 (1995) · Zbl 0831.05044
[155] Hoffman, A. J., \(- 1 - \sqrt{2} \)? Combinatorial structures and their applications, (Guy, R.; Hanani, H.; Sauer, N.; Schönheim, J., Proc. Calgary Intern. Conf. on Combinatorial Structures and Their Applications Held at the Univ. of Calgary. Proc. Calgary Intern. Conf. on Combinatorial Structures and Their Applications Held at the Univ. of Calgary, June 1969 (1970), Gordon and Breach Inc.: Gordon and Breach Inc. New York, London, Paris), 173-176 · Zbl 0262.05133
[156] Hoffman, A. J., On limit points of the least eigenvalue of a graph, Ars Combin., 3, 3-14 (1977) · Zbl 0445.05067
[157] Hoffman, A. J., On graphs whose least eigenvalue exceeds \(- 1 - \sqrt{2} \), Linear Algebra Appl., 16, 153-165 (1977) · Zbl 0354.05048
[158] McKee, J.; Rowlinson, P.; Smyth, C., Salem numbers and Pisot numbers from stars, (Gyory, K.; Iwaniec, H.; Urbanowicz, J., Number Theory in Progress, Proc. International Conference on Number Theory, Part 1: Diophantine Problems and Polynomials. Number Theory in Progress, Proc. International Conference on Number Theory, Part 1: Diophantine Problems and Polynomials, Zakopane, Poland, 1997 (1999), Walter De Gruyter: Walter De Gruyter Berlin), 309-319 · Zbl 0931.11041
[159] Neumaier, A., The second largest eigenvalue of a tree, Linear Algebra Appl., 46, 9-25 (1982) · Zbl 0495.05044
[160] Radosavljević, Z.; Simić, S., Which bicyclic graphs are reflexive?, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 7, 90-104 (1996) · Zbl 0941.05044
[161] Reed, B. A., \(ω\), Δ, and \(χ\), J. Graph Theory, 27, 177-212 (1998) · Zbl 0980.05026
[162] Sciriha, I., The two classes of singular line graphs of trees, Rend. Sem. Mat. Messina Ser. II, 5, 167-180 (1999) · Zbl 0959.05073
[163] Torgašev, A., A note on infinite generalized line graphs, (Cvetković, D.; Gutman, I.; Pisanski, T.; Tošić, R., Graph Theory, Proc. Fourth Yugoslav Sem. Graph Theory. Graph Theory, Proc. Fourth Yugoslav Sem. Graph Theory, Novi Sad, 15-16 April 1983 (1984), Inst. Math. Novi Sad), 291-297 · Zbl 0541.05042
[164] Torgašev, A., Infinite graphs with the least limiting eigenvalue greater than −2, Linear Algebra Appl., 82, 133-141 (1986) · Zbl 0611.05040
[165] Vijayakumar, G. R., Signed graphs represented by \(D_\infty \), European J. Combin., 8, 103-111 (1987) · Zbl 0678.05058
[166] Woo, R.; Neumaier, A., On graphs whose smallest eigenvalue is at least \(- 1 - \sqrt{2} \), Linear Algebra Appl., 226-228, 577-591 (1995) · Zbl 0832.05076
[167] Yong, X., On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl., 295, 73-80 (1999) · Zbl 0931.05056
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.