New Bounds for the ? -Indices of Graphs

Eber Lenes, Exequiel Mallea-Zepeda and Jonnathan Rodríguez
Additional contact information
Eber Lenes: Área de Ciencias Básicas Exactas, Grupo de Investigación Deartica, Universidad del Sinú, Cartagena 130001, Colombia
Exequiel Mallea-Zepeda: Departamento de Matemática, Universidad de Tarapacá, 1000000 Arica, Chile
Jonnathan Rodríguez: Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Av. Angamos 601, 1240000 Antofagasta, Chile

Mathematics, 2020, vol. 8, issue 10, 1-12

Abstract: Let G be a graph, for any real 0 ≤ α ≤ 1 , Nikiforov defines the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where A ( G ) and D ( G ) are the adjacency matrix and diagonal matrix of degrees of the vertices of G . This paper presents some extremal results about the spectral radius ρ α ( G ) of the matrix A α ( G ) . In particular, we give a lower bound on the spectral radius ρ α ( G ) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρ α ( G ) in terms of order and minimal degree. Furthermore, for n > l > 0 and 1 ≤ p ≤ ⌊ n − l 2 ⌋ , let G p ≅ K l ∨ ( K p ∪ K n − p − l ) be the graph obtained from the graphs K l and K p ∪ K n − p − l and edges connecting each vertex of K l with every vertex of K p ∪ K n − p − l . We prove that ρ α ( G p + 1 ) < ρ α ( G p ) for 1 ≤ p ≤ ⌊ n − l 2 ⌋ − 1 .

Keywords: spectral radius; minimal degree; independence number; ?-adjacency matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/10/1668/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/10/1668/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:10:p:1668-:d:420782

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().