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Bounds on the \(ABC\) spectral radius of a tree. (English) Zbl 1468.05030

Summary: Let \(G\) be a simple connected graph with vertex set \(\{1,2,\dots,n\}\) and \(d_i\) denote the degree of vertex \(i\) in \(G\). The \(ABC\) matrix of \(G\), recently introduced by Estrada, is the square matrix whose \(ij\)th entry is \(\sqrt{\frac{d_i+d_j-2}{d_id_j}}\); if \(i\) and \(j\) are adjacent, and zero; otherwise. The entries in \(ABC\) matrix represent the probability of visiting a nearest neighbor edge from one side or the other of a given edge in a graph. In this article, we provide bounds on \(ABC\) spectral radius of \(G\) in terms of the number of vertices in \(G\). The trees with maximum and minimum \(ABC\) spectral radius are characterized. Also, in the class of trees on \(n\) vertices, we obtain the trees having first four values of \(ABC\) spectral radius and subsequently derive a better upper bound.

MSC:

05C05 Trees
05C35 Extremal problems in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
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[1] A. Chang and Q. Huang, Ordering trees by their largest eigenvalues, Linear Algebra Appl. 370 (2003), 175-184. · Zbl 1030.05029
[2] X. Chen, On ABC eigenvalues and ABC energy, Linear Algebra Appl. 544 (2018), 141-157. · Zbl 1388.05112
[3] J. Chen, J. Liu, and X. Guo, Some upper bounds for the atom-bond connectivity index of graphs, Appl. Math. Lett., 25 (2012), 1077-1081. · Zbl 1246.05091
[4] D. Cvetković and P. Rowlinson, The largest eigenvalue of a graph: A survey, Linear Multi-linear Algebra, 544 (1990), 141-157. · Zbl 0744.05031
[5] K.C. Das, Atom-bond connectivity index of graphs, Discrete Appl. Math., 158 (2010), 1181-1188. · Zbl 1230.05184
[6] E. Estrada, The ABC matrix, J. Math. Chem., 55 (2017), 1021-1033. · Zbl 1380.92097
[7] E. Estrada, L. Torres, L. Rodríguez, and I. Gutman, An atom-bond connectivity index: mod-elling the enthalpy of formation of alkanes, Indian J. Chem., 37(A) (1998), 849-855.
[8] G.H. Fath-Tabar, B. Vaez-Zadeh, A.R. Ashrafi, and A. Graovac, Some inequalities for the atom-bond connectivity index of graph operations, Discrete Appl. Math., 159 (2011), 1323-1330. · Zbl 1223.05154
[9] B. Furtula, A. Graovac, and D. Vukičević, Atom-bond connectivity index of trees, Discrete Appl. Math., 157 (2009), 2828-2835. · Zbl 1209.05252
[10] B. Furtula, I. Gutman, and M. Ivanović, D. Vukičević, Computer search for trees with mini-mal ABC index, Appl. Math. Comput., 219 (2012), 768-772. · Zbl 1285.05028
[11] L. Gan, H. Hou, and B. Liu, Some results on atom-bond connectivity index of graphs, MATCH Commun. Math. Comput. Chem., 66 (2011), 669-680. · Zbl 1265.05576
[12] I. Gutman and B. Furtula, Trees with smallest atom-bond connectivity index, MATCH Com-mun. Math. Comput. Chem., 68 (2012), 131-136. · Zbl 1289.05047
[13] I. Gutman, B. Furtula, and M. Ivanović, Notes on trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem., 67 (2012), 467-482. · Zbl 1289.05063
[14] I. Gutman, J. Tošović, S. Radenković, and S. Marković, On atom-bond connectivity index and its chemical applicability, Indian J. Chem., 51(A) (2012), 690-694.
[15] M. Hofmeister, On the two largest eigenvalues of trees, Linear Algebra Appl., 260 (1997), 43-59. · Zbl 0876.05068
[16] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Uni. Press, New York, (2013). · Zbl 1267.15001
[17] W. Lin, X. Lin, T. Gao, and X. Wu, Proving a conjecture of Gutman concerning trees with minimal ABC index, MATCH Commun. Math. Comput. Chem., 69 (2013) 549-557. · Zbl 1299.05038
[18] L. Lovász, Random walks on graphs: A Survey, Combinatorics Paul Erdős is Eighty, 2 (1993) 1-46.
[19] R. Xing, B. Zhou, and Z. Du, Further results on atom-bond connectivity index of trees, Discrete Appl. Math., 158 (2010), 1536-1545. · Zbl 1216.05161
[20] R. Xing, B. Zhou, and F. Dong, On atom-bond connectivity index of connected graphs, Dis-crete Appl. Math., 159 (2011) 1617-1630. · Zbl 1228.05199
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