Majorization under constraints and bounds on the second Zagreb index

Discrete Applied Mathematics

Palacios

et al. 2015

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Making use of a majorization technique for a suitable class of graphs, we derive upper and lower bounds for some topological indices depending on the degree sequence over all vertices, namely the first general Zagreb index and the first multiplicative Zagreb index. Specifically, after characterizing c−cyclic graphs (0 ≤ c ≤ 6) as those whose degree sequence belongs to particular subsets of R n , we identify the maximal and minimal vectors of these subsets with respect to the majorization order. This technique allows us to determine lower and upper bounds of the above indices recovering those existing in the literature as well obtaining new ones.

Section: Notations and Preliminaries Results On Majorizationmentioning

confidence: 99%

Section: Notations and Preliminaries Results On Majorizationmentioning

confidence: 99%

New bounds of degree-based topological indices for some classes of c-cyclic graphs

Bianchi

Discrete Applied Mathematics

Palacios

et al. 2015

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“…It is noteworthy to state that the results in Theorem 2 are tighter than those in Theorem 1 (for more details see [3] and [4]).…”

Section: Notations and Preliminariesmentioning

confidence: 88%

“…Since λ 1 ≥ Q ≥ P , we provide bounds (1) and (2) better than in [7] (see [3] and [4] for more theoretical details).…”

Section: New Bounds For Normalized Laplacian Eigenvaluesmentioning

confidence: 95%

New bounds for the sum of powers of normalized Laplacian eigenvalues of graphs

Clemente

2016

Ars Math. Contemp.

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For a simple and connected graph, a new graph invariant s * α (G), defined as the sum of α powers of the eigenvalues of the normalized Laplacian matrix, has been introduced by Bozkurt and Bozkurt in [7]. Lower and upper bounds have been proposed by the authors. In this paper, we localize the eigenvalues of the normalized Laplacian matrix by adapting a theoretical method, proposed in Bianchi and Torriero ([5]), based on majorization techniques. Through this approach we derive upper and lower bounds of s * α (G). Some numerical examples show how sharper results can be obtained with respect to those existing in literature.

“…The contribution of this paper is along those lines: we derive, through a methodology based on majorization techniques (see [15–17] and [18]), new bounds on the median eigenvalues of the normalized Laplacian matrix. Consequently, given the relation between the normalized Laplacian matrix and the adjacency matrix eigenvalues, we provide some new bounds for the HL -index.…”

Section: Introductionmentioning

confidence: 99%

Bounding the HL-index of a graph: a majorization approach

Clemente

2016

J Inequal Appl

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In mathematical chemistry, the median eigenvalues of the adjacency matrix of a molecular graph are strictly related to orbital energies and molecular orbitals. In this regard, the difference between the occupied orbital of highest energy (HOMO) and the unoccupied orbital of lowest energy (LUMO) has been investigated (see Fowler and Pisansky in Acta Chim. Slov. 57:513-517, 2010). Motivated by the HOMO-LUMO separation problem, Jaklič et al. in (Ars Math. Contemp. 5:99-115, 2012) proposed the notion of HL-index that measures how large in absolute value are the median eigenvalues of the adjacency matrix. Several bounds for this index have been provided in the literature. The aim of the paper is to derive alternative inequalities to bound the HL-index. By applying majorization techniques and making use of some known relations, we derive new and sharper upper bounds for this index. Analytical and numerical results show the performance of these bounds on different classes of graphs.