Energy of graphs with respect to generalized distance matrix: Extremal results and bounds

Abdollah Alhevaz (), Maryam Baghipur (), Kinkar Chandra Das () and Yilun Shang ()
Additional contact information
Abdollah Alhevaz: Shahrood University of Technology
Maryam Baghipur: Shahrood University of Technology
Kinkar Chandra Das: Sungkyunkwan University
Yilun Shang: Northumbria University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 4, 1478-1494

Abstract: Abstract Let G be a simple connected graph of order n. Denote by D(G) the distance matrix of G and by Tr(G) the diagonal matrix of its vertex transmissions. For $$0\le \alpha \le 1$$ 0 ≤ α ≤ 1 , the generalized distance matrix $$D_{\alpha }(G)$$ D α ( G ) of G is defined as $$D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)$$ D α ( G ) = α T r ( G ) + ( 1 - α ) D ( G ) . The generalized distance energy of a graph G (energy of G with respect to the generalized distance matrix) is defined as $$E^{D_{\alpha }}(G)=\sum _{i=1}^{n}\left| \partial _i-\frac{2\alpha W(G)}{n}\right| ,$$ E D α ( G ) = ∑ i = 1 n ∂ i - 2 α W ( G ) n , where W(G) is the transmission (also called the Wiener index) of a graph G and $$\partial _{1}\ge \partial _{2}\ge \cdots \ge \partial _{n}$$ ∂ 1 ≥ ∂ 2 ≥ ⋯ ≥ ∂ n are the eigenvalues of $$D_{\alpha }(G)$$ D α ( G ) . In this paper, we establish new upper and lower bounds for $$E^{D_{\alpha }}(G)$$ E D α ( G ) in terms of various graph invariants, and we characterize the extremal graphs for which these bounds are attained.

Keywords: Generalized distance matrix; Generalized distance energy; Bound; Extremal graph (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-025-00846-x 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:56:y:2025:i:4:d:10.1007_s13226-025-00846-x

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

DOI: 10.1007/s13226-025-00846-x

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 ().