Connected graphs having maximum signless Laplacian spectral radius
Abstract
Let ρ(G) denote the signless Laplacian spectral radius of a graph G, and let 𝒞n,m be the family of connected graphs of order n and size m. A graph is Q-maximal if it maximizes ρ(G) over this family. Motivated by a question of Chang and Tam (2011), we study this extremal problem using scalar identities and coronal methods for threshold graphs. From the Perron equations, we derive the identity SG(λ)=Ta,b(λ) at λ=ρ(G), where a and b are the total numbers of zero-vertices and one-vertices. This yields a one-point comparison principle: for a connected threshold graph H with the same block totals, SH(z)>Ta,b(z) at some z≥ n implies ρ(H)>z, allowing spectral comparison without estimating individual Perron coordinates. We prove that every Q-maximal threshold graph has character at most three, so its vertex-sequence contains at most three zero--one block pairs. We also show that the quasi-star graph is Q-maximal whenever n≥5 and n−1≤ m<(n2+10n−8)/8. For certain dense parameter pairs, we construct connected threshold graphs with signless Laplacian spectral radii strictly larger than those of both the quasi-star and the connected quasi-complete graph. Thus the corresponding two-candidate assertion for connected graphs fails in general.
Related
- Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs
- A solution to a conjecture on the signless Laplacian spectral radius for t-color-critical graphs
- Signless Laplacian spectral conditions for rainbow matchings in a collection of bipartite graphs
- A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound
- The maximum spectral radius of connected graphs with prescribed order and size
BibTeX
@misc{wang2026connected,
title = {{Connected graphs having maximum signless Laplacian spectral radius}},
author = {Di-Jian Wang and Hong-Yan Ge and Shi-Cai Gong and Bo-Jun Yuan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07429},
eprint = {2611.07429},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07429},
}
RIS
TY - UNPB
TI - Connected graphs having maximum signless Laplacian spectral radius
AU - Di-Jian Wang
AU - Hong-Yan Ge
AU - Shi-Cai Gong
AU - Bo-Jun Yuan
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07429
UR - https://arxiv.org/abs/2611.07429
AB - Let ρ(G) denote the signless Laplacian spectral radius of a graph G, and let 𝒞_n,m be the family of connected graphs of order n and size m. A graph is Q-maximal if it maximizes ρ(G) over this family. Motivated by a question of Chang and Tam (2011), we study this extremal problem using scalar identities and coronal methods for threshold graphs. From the Perron equations, we derive the identity S_G(λ)=T_a,b(λ) at λ=ρ(G), where a and b are the total numbers of zero-vertices and one-vertices. This yields a one-point comparison principle: for a connected threshold graph H with the same block totals, S_H(z)>T_a,b(z) at some z≥ n implies ρ(H)>z, allowing spectral comparison without estimating individual Perron coordinates. We prove that every Q-maximal threshold graph has character at most three, so its vertex-sequence contains at most three zero--one block pairs. We also show that the quasi-star graph is Q-maximal whenever n≥5 and n−1≤ m<(n^2+10n−8)/8. For certain dense parameter pairs, we construct connected threshold graphs with signless Laplacian spectral radii strictly larger than those of both the quasi-star and the connected quasi-complete graph. Thus the corresponding two-candidate assertion for connected graphs fails in general.
ER -