On squared distance matrix of complete multipartite graphs

Joyentanuj Das () and Sumit Mohanty ()
Additional contact information
Joyentanuj Das: National Sun Yat-sen University
Sumit Mohanty: IIM Ranchi

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 517-537

Abstract: Abstract Let $$G = K_{n_1,n_2,\cdots ,n_t}$$ G = K n 1 , n 2 , ⋯ , n t be a complete t-partite graph on $$n=\sum _{i=1}^t n_i$$ n = ∑ i = 1 t n i vertices. The distance between vertices i and j in G, denoted by $$d_{ij}$$ d ij is defined to be the length of the shortest path between i and j. The squared distance matrix $$\Delta (G)$$ Δ ( G ) of G is the $$n\times n$$ n × n matrix with $$(i,j)^{th}$$ ( i , j ) th entry equal to 0 if $$i = j$$ i = j and equal to $$d_{ij}^2$$ d ij 2 if $$i \ne j$$ i ≠ j . We define the squared distance energy $$E_{\Delta }(G)$$ E Δ ( G ) of G to be the sum of the absolute values of its eigenvalues. We determine the inertia of $$\Delta (G)$$ Δ ( G ) and compute the squared distance energy $$E_{\Delta }(G)$$ E Δ ( G ) . More precisely, we prove that if $$n_i \ge 2$$ n i ≥ 2 for $$1\le i \le t$$ 1 ≤ i ≤ t , then $$ E_{\Delta }(G)=8(n-t)$$ E Δ ( G ) = 8 ( n - t ) and if $$ h= |\{i: n_i=1\}|\ge 1$$ h = | { i : n i = 1 } | ≥ 1 , then $$\begin{aligned} 8(n-t)+2(h-1) \le E_{\Delta }(G)

Keywords: Complete t-partite graphs; Squared distance matrix; Inertia; Energy; Spectral radius; 05C12; 05C50 (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-023-00386-2 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:55:y:2024:i:2:d:10.1007_s13226-023-00386-2

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

DOI: 10.1007/s13226-023-00386-2

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 ().