On Schatten p-norm of the distance matrices of graphs

Bilal Ahmad Rather ()
Additional contact information
Bilal Ahmad Rather: University of Kashmir

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 4, 1012-1024

Abstract: Abstract For a connected simple graph G, the generalized distance matrix is defined by $$ D_{\alpha }:= \alpha Tr(G)+(1-\alpha ) D(G), ~0\le \alpha \le 1 $$ D α : = α T r ( G ) + ( 1 - α ) D ( G ) , 0 ≤ α ≤ 1 , where Tr(G) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix. For particular values of $$ \alpha $$ α , we obtain the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix and other uncountable distance based matrices. Let $$ \partial _{1}\ge \partial _{2}\ge \dots \ge \partial _{n} $$ ∂ 1 ≥ ∂ 2 ≥ ⋯ ≥ ∂ n be the $$ D_{\alpha } $$ D α eigenvalues of G and $$ p\ge 2 $$ p ≥ 2 a real number, the Schatten p-norm is the p-th root of the sum of p-th powers of eigenvalues of $$ D_{\alpha }, ~\alpha \in [\frac{1}{2},1] $$ D α , α ∈ [ 1 2 , 1 ] , that is, $$ \Vert D_{\alpha }\Vert _{p}^{p} =\partial _{1}^{p}+\partial _{2}^{p}+\dots +\partial _{n}^{p}$$ ‖ D α ‖ p p = ∂ 1 p + ∂ 2 p + ⋯ + ∂ n p . In this paper, we obtain various bounds for $$ \Vert D_{\alpha }\Vert _{p}^{p} $$ ‖ D α ‖ p p in terms of different graph parameters and characterize the corresponding extremal graphs.

Keywords: Distance matrix; Distance Laplacian matrix; Distance Signless Laplacian matrix; $$ D_\alpha $$ D α matrix; Schatten p-norm; Ky Fan k norm; 05C50; 05C12; 15A18. (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-022-00317-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

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:spr:indpam:v:54:y:2023:i:4:d:10.1007_s13226-022-00317-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-022-00317-7

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().