Laplacian energy and first Zagreb index of Laplacian integral graphs

[1] de Abreu NMM, Vinagre CTM, Bonifácio AS, Gutman I. The Laplacian energy of some Laplacian integral graphs. MATCH Communications in Mathematical and in Computer Chemistry 2008; 60 (2): 447–460. Search in Google Scholar

[2] Ashrafi AR, Došlić T, Hamzeh A. The Zagreb coindices of graph operations. Discrete Applied Mathematics 2010; 158 (15): 1571–1578. doi: 10.1016/j.dam.2010.05.017.10.1016/j.dam.2010.05.017 Search in Google Scholar

[3] Balaban AT, Motoc I, Bonchev D, Mekenyan O. Topological indices for structure-activity correlations. Topics in Current Chemistry 1983; 114: 21–55. doi: 10.1007/BFB0111212.10.1007/BFb0111212 Search in Google Scholar

[4] Balakrishnan R. The energy of a graph. Linear Algebra and its Applications 2004; 387: 287–295. doi: 10.1016/j.laa.2004.02.038.10.1016/j.laa.2004.02.038 Search in Google Scholar

[5] Balakrishnan R, Kanna R. A textbook of graph theory. Springer Science & Business Media, 2012. doi: org/10.1007/978-1-4614-4529-6. Search in Google Scholar

[6] Cvetković D, Rowlinson P, Simić S. An introduction to the theory of graph spectra. London Mathematical Society Student Texts 2010; 75: xii+364. doi: 10.1017/CBO9780511801518.10.1017/CBO9780511801518 Search in Google Scholar

[7] Das KC. On comparing Zagreb indices of graphs. MATCH Communications in Mathematical and in Computer Chemistry 2010; 63 (2): 433–440. Search in Google Scholar

[8] Das KC, Gutman I. Some properties of the second Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry 2004; 52 (1): 103–112. Search in Google Scholar

[9] Das KC, Gutman I, Zhou B. New upper bounds on Zagreb indices. Journal of Mathematical Chemistry 2009; 46 (2): 514–521.10.1007/s10910-008-9475-3 Search in Google Scholar

[10] Das KC, Lee SG, Cheon GS. On the conjecture for certain Laplacian integral spectrum of graphs. Journal of Graph Theory 2010; 63 (2): 106–113. doi: 10.1002/jgt.20412.10.1002/jgt.20412 Search in Google Scholar

[11] Das KC, Mojallal SA. On Laplacian energy of graphs. Discrete Mathematics 2014; 325: 52–64.doi: 10.1016/j.disc.2014.02.017.10.1016/j.disc.2014.02.017 Search in Google Scholar

[12] Das KC, Mojallal SA, Gutman I. On Laplacian energy in terms of graph invariants. Applied Mathematics and Computation 2015; 268: 83–92. doi: 10.1016/j.amc.2015.06.06410.1016/j.amc.2015.06.064 Search in Google Scholar

[13] Das KC, Mojallal SA, Gutman I. On energy and Laplacian energy of bipartite graphs. Applied Mathematics and Computation 2016; 273: 759–766. doi: 10.1016/j.amc.2015.10.047.10.1016/j.amc.2015.10.047 Search in Google Scholar

[14] Del-Vecchio RR, Pereira GB, Vinagre C.T.M. Constructing pairs of Laplacian equienergetic threshold graphs. Matemtica Contempornea 2016; 44 :1–10. Search in Google Scholar

[15] Deng B, Li X. More on L-borderenergetic graphs. MATCH Communications in Mathematical and in Computer Chemistry 2017; 77 (1): 115–127. Search in Google Scholar

[16] Deng B, Li X. On L-borderenergetic graphs with maximum degree at most 4. MATCH Communications in Mathematical and in Computer Chemistry 2018; 79(2): 303–310. Search in Google Scholar

[17] Fallat SM, Kirkland SJ, Molitierno JJ, Neumann M. On graphs whose Laplacian matrices have distinct integer eigenvalues. Journal of Graph Theory 2005; 50 (2): 162–174. doi: 10.1002/jgt.20102.10.1002/jgt.20102 Search in Google Scholar

[18] Fritscher E, Hoppen C, Trevisan V. Unicyclic graphs with equal Laplacian energy. Linear and Multilinear Algebra 2014; 62 (2): 180–194. doi: 10.1080/03081087.2013.766395.10.1080/03081087.2013.766395 Search in Google Scholar

[19] Ganie HA, Shariefuddin P, Iványi A. Energy, Laplacian energy of double graphs and new families of equienergetic graphs. Acta Universitatis Sapientiae, Informatica 2014; 6(1): 89–116. doi: 10.2478/ausi-2014-0020.10.2478/ausi-2014-0020 Search in Google Scholar

[20] Gibbs RA. Self-complementary graphs. Journal of Combinatorial Theory, Series B 1974; 16: 106–123. doi: 10.1016/0095-8956(74)90053-7.10.1016/0095-8956(74)90053-7 Search in Google Scholar

[21] Goldberger A, Neumann M. On a conjecture on a Laplacian matrix with distinct integral spectrum. Journal of Graph Theory 2013; 72 (2): 178–208. doi: 10.1002/jgt.21638.10.1002/jgt.21638 Search in Google Scholar

[22] Gong S, Li X, Xu G, Gutman I, Furtula B. Borderenergetic graphs. MATCH Communications in Mathematical and in Computer Chemistry 2015; 74 (2): 321–332. Search in Google Scholar

[23] Gutman I. The energy of a graphs. Berichte Mathematisch- Statistische Sektion Forschungszentrum Graz 1978; 103: 1–22. Search in Google Scholar

[24] Gutman I. The energy of a graph: old and new results. Algebraic Combinatorics and Application 2001; 196–211. doi: 10.1007/978-3-642-59448-9_13.10.1007/978-3-642-59448-9_13 Search in Google Scholar

[25] Gutman I, de Abreu N.M.M, Vinagre C.T.M, Bonifácio AS, Radenković S. Relation between energy and Laplacian energy. MATCH Communications in Mathematical and in Computer Chemistry 2008; 59 (2): 343–354. Search in Google Scholar

[26] Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Communications in Mathematical and in Computer Chemistry 2004; 50: 83–92. Search in Google Scholar

[27] Gutman I, Ruščić B, Trinajstić N, Wilcox CF. Graph theory and molecular orbitals. XII. Acyclic polyenes. The Journal of Chemical Physics 1975; 62 (9): 3399–3405. doi: 10.1063/1.430994.10.1063/1.430994 Search in Google Scholar

[28] Gutman I, Trinajstić N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chemical Physics Letters 1972; 17 (4): 535–538.doi: 10.1016/0009-2614(72)85099-1.10.1016/0009-2614(72)85099-1 Search in Google Scholar

[29] Gutman I, Zhou B. Laplacian energy of a graph. Linear Algebra and its Applications 2006; 414 (1):29–37. doi: 10.1016/j.laa.2005.09.008.10.1016/j.laa.2005.09.008 Search in Google Scholar

[30] Horoldagva B, Das KC. On comparing Zagreb indices of graphs. Hacettepe Journal of Mathematics and Statistics 2012; 41 (2): 223-230. Search in Google Scholar

[31] Hosamani M, Basavanagoud B. New upper bounds for the first Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry 2015; 74(1):97–101. Search in Google Scholar

[32] Ilić A, Stevanović D. On comparing Zagreb indices. MATCH Communications in Mathematical and in Computer Chemistry 2009;62 (4): 681–687. Search in Google Scholar

[33] Kel’mans AK. The number of trees of a graph I. Automat. Remote Control 1966; 26: 2118–2129 (1966). Search in Google Scholar

[34] Khalifeh MH, Yousefi-Azari H, Ashrafi A. The first and second Zagreb indices of some graph operations. Discrete applied mathematics 2009; 157 (4): 804–811. doi: 10.1016/j.dam.2008.06.015.10.1016/j.dam.2008.06.015 Search in Google Scholar

[35] Liu J, Liu B, Radenković S, Gutman I. Minimal LEL-equienergetic graphs. MATCH Communications in Mathematical and in Computer Chemistry 2009; 61: 471–478. Search in Google Scholar

[36] Merris R. Laplacian matrices of graphs: a survey. Linear Algebra and its Applications 1994; 197–198: 143–176. doi: 10.1016/0024-3795(94)90486-3.10.1016/0024-3795(94)90486-3 Search in Google Scholar

[37] Merris R. Degree maximal graphs are Laplacian integral. Linear Algebra and its Applications 1994; 199: 381–389. doi: 10.1016/0024-3795(94)90361-1.10.1016/0024-3795(94)90361-1 Search in Google Scholar

[38] Merris R. Graph theory. Wiley-Interscience Series in Discrete Mathematics and Optimization 2001; 50 (2): xiv+237. doi: 10.1002/9781118033043.scard.10.1002/9781118033043.scard Search in Google Scholar

[39] Milovanović IZ, Matejić M, Milošević P, Milovanović E, Ali A. A note on some lower bounds of the Laplacian energy of a graph. Transactions on Combinatorics 2019; 8 (2): 13–19. Search in Google Scholar

[40] Mohar B. The Laplacian spectrum of graphs. Graph theory, combinatorics, and applications 1991; 2: 871–898. Search in Google Scholar

[41] Neumann M, Pati S. The Laplacian spectra of graphs with a tree structure. Linear and Multilinear Algebra 2009; 57 (3): 267–291. doi: 10.1080/03081080701688101.10.1080/03081080701688101 Search in Google Scholar

[42] Nikolić S, Kovačević G, Miličević A, Trinajstić N. The Zagreb indices 30 years after. Croatica Chemica Acta 2003; 76 (2): 113–124. Search in Google Scholar

[43] Robbiano M, Martins EA, Jiménez R, San Martín B. Upper bounds on the Laplacian energy of some graphs. MATCH Communications in Mathematical and in Computer Chemistry 2010; 64 (1): 97–110. Search in Google Scholar

[44] Stevanović D. Large sets of noncospectral graphs with equal Laplacian energy. MATCH Communications in Mathematical and in Computer Chemistry 2009; 61 (2): 463. Search in Google Scholar

[45] Stevanović D, Stanković I, Milošević M. More on the relation between energy and Laplacian energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry 2009; 61 (2): 395–401. Search in Google Scholar

[46] Su G, Xiong L, Xu L. The Nordhaus–Gaddum-type inequalities for the Zagreb index and co-index of graphs. Applied Mathematics Letters 2012; 25 (11): 1701–1707.10.1016/j.aml.2012.01.041 Search in Google Scholar

[47] Tao Q, Hou Y. A computer search for the L-borderenergetic graph. MATCH Communications in Mathematical and in Computer Chemistry 2017; 77 (3): 595–606. Search in Google Scholar

[48] Todeschini R, Consonni V. Handbook of Molecular Descriptors. John Wiley & Son, 2008.10.1007/978-1-4020-9783-6_3 Search in Google Scholar

[49] Tura F. L-borderenergetic graphs and normalized Laplacian energy. MATCH Communications in Mathematical and in Computer Chemistry 2017; 77 (3): 617–624. Search in Google Scholar

[50] Tura F. L-borderenergetic graphs. MATCH Communications in Mathematical and in Computer Chemistry 2017; 77: 37–44. Search in Google Scholar

[51] Wang M, Hua H. More on Zagreb coindices of composite graphs. International Mathematical Forum 2012; 7: 669–673. Search in Google Scholar

[52] Zhou B. More on energy and Laplacian energy. MATCH Communications in Mathematical and in Computer Chemistry 2010; 64 (1): 75–84. Search in Google Scholar

[53] Zhou B, Gutman I, Aleksić T. A note on Laplacian energy of graphs. MATCH Communications in Mathematical and in Computer Chemistry 2008; 60 (2): 441–446. Search in Google Scholar