×

Maximum first Zagreb index of orientations of unicyclic graphs with given matching number. (English) Zbl 1510.05045

Summary: Let \(D = ( V , A )\) be a digraphs without isolated vertices. The first Zagreb index of a digraph \(D\) is defined as a summation over all arcs, \( M_1 ( D ) = \frac{1}{2} \sum_{u v \in A} ( d_u^+ + d_v^- )\), where \(d_u^+\) (resp. \(d_u^-\)) denotes the out-degree (resp. in-degree) of the vertex \(u\). In this paper, we give the maximal values and maximal digraphs of first Zagreb index over the set of all orientations of unicyclic graphs with \(n\) vertices and matching number \(m ( 2 \leq m \leq \lfloor \frac{n}{2} \rfloor )\).

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C07 Vertex degrees
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Gutman, I.; Trinajstić, N., Graph theory and molecular orbitals. Total \(\pi \)-electronenergy of alternant hydrocarbons, Chem. Phys. Lett., 17, 535-538 (1972)
[2] Gutman, I.; Ruščić, B.; Trinajstić, N.; Wilcox, C. F., Graph theory and molecularorbitals. XII. Acyclic polyenes, J. Chem. Phys., 62, 3399-3405 (1975)
[3] Ali, A.; Gutman, I.; Milovanović, E.; Milovanović, I., Sum of powers of the degrees of graphs: extremal results and bounds, MATCH Commun. Math. Comput. Chem., 80, 5-84 (2018) · Zbl 1468.05288
[4] Borovićanin, B.; Das, K. C.; Furtula, B.; Gutman, I., Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem., 78, 17-100 (2017) · Zbl 1471.92406
[5] Das, K. C., Sharp bounds for the sum of the squares of degrees of a graph, Kragujevac J. Math., 25, 31-49 (2003) · Zbl 1051.05034
[6] Das, K. C., Maximizing the sum of the squares of the degrees of a graph, Discr. Math., 285, 57-66 (2004) · Zbl 1051.05033
[7] Das, K. C., On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 63, 433-440 (2010) · Zbl 1265.05114
[8] Das, K. C.; Gutman, I.; Zhou, B., New upper bounds on Zagreb indices, J. Math. Chem., 46, 514-521 (2009) · Zbl 1200.92048
[9] Zhou, B., Remarks on Zagreb indices, MATCH Commun. Math. Comput. Chem., 57, 591-596 (2007) · Zbl 1141.05027
[10] Yu, A.; Tian, F., On the spectral radius of unicyclic graphs, MATCH Commun. Math. Comput. Chem., 51, 97-109 (2004) · Zbl 1053.05085
[11] Nikolić, S.; Kovačević, G.; Miličević, A.; Trinajstić, N., The Zagreb indices 30 years after, Croat. Chem. Acta., 76, 2, 113-124 (2003)
[12] Monsalve, J.; Rada, J., Vertex-degree based topological indices of digraphs, Discrete Appl. Math., 295, 13-24 (2021) · Zbl 1460.05046
[13] Monsalve, J.; Rada, J., Oriented bipartite graphs with minimal trace norm, Linear Multilinear A., 67, 6, 1121-1131 (2019) · Zbl 1411.05172
[14] Monsalve, J.; Rada, J., Sharp upper and lower bounds of VDB topological indices of digraphs, Symmetry, 13, 1903-1904 (2021)
[15] Luo, W.; Zhou, B., On the irregularity of trees and unicyclic graphs with given matching number, Util. Math., 83, 141-148 (2010) · Zbl 1242.05223
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.