Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs
Abstract
Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs ℱ, a graph is called ℱ-free if it contains no member of ℱ as a subgraph. The theta graph θ(l1,…,lk) consists of k internally disjoint paths of lengths l1,…,lk with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all {C3,C4}-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all {θ(1,2,2),θ(1,2,3)}-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.
Related
- The signless Laplacian spectral Turán problems for disjoint triangles
- Proofs of Li, Wang and Guo's conjectures on (signless) Laplacian spectral radius
- 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
- The Distance Laplacian and Distance Signless Laplacian Spectra of 𝒞-Graphs
BibTeX
@misc{basunia2026characterization,
title = {{Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs}},
author = {Mainak Basunia and Pratima Panigrahi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17154},
eprint = {2609.17154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17154},
}
RIS
TY - UNPB
TI - Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs
AU - Mainak Basunia
AU - Pratima Panigrahi
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17154
UR - https://arxiv.org/abs/2609.17154
AB - Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs ℱ, a graph is called ℱ-free if it contains no member of ℱ as a subgraph. The theta graph θ(l_1,…,l_k) consists of k internally disjoint paths of lengths l_1,…,l_k with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all {C_3,C_4}-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all {θ(1,2,2),θ(1,2,3)}-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.
ER -