Let $ H $ be a spanning subgraph of $ G $. If every vertex $ v\in V(G) $ satisfies $ a\leq d_{H}(v)\leq b $, then $ H $ is called an $ [a, b] $-factor of $ G $. In 2005, Brouwer and Haemers pioneered the spectral approach for investigating $ 1 $-factors in regular graphs. Since this work, adjacency eigenvalue conditions for $ [a, b] $-factors have been extensively studied. In this paper, we provided some conditions based on the distance spectral radius that ensured the existence of an $ [a, b] $-factor in a connected graph and a balanced bipartite graph, respectively.
Citation: Jihan Xin, Huixin Kang, Qi He, Dandan Fan. Distance spectral radius and $ [a, b] $-factor of graphs[J]. AIMS Mathematics, 2026, 11(5): 12360-12372. doi: 10.3934/math.2026507
Let $ H $ be a spanning subgraph of $ G $. If every vertex $ v\in V(G) $ satisfies $ a\leq d_{H}(v)\leq b $, then $ H $ is called an $ [a, b] $-factor of $ G $. In 2005, Brouwer and Haemers pioneered the spectral approach for investigating $ 1 $-factors in regular graphs. Since this work, adjacency eigenvalue conditions for $ [a, b] $-factors have been extensively studied. In this paper, we provided some conditions based on the distance spectral radius that ensured the existence of an $ [a, b] $-factor in a connected graph and a balanced bipartite graph, respectively.
| [1] |
J. Petersen, Die theorie der regulären graphs, Acta Math., 15 (1891), 193–220. https://doi.org/10.1007/BF02392606 doi: 10.1007/BF02392606
|
| [2] |
A. Brouwer, W. Haemers, Eigenvalues and perfect matchings, Linear Algebra Appl., 395 (2005), 155–162. https://doi.org/10.1016/j.laa.2004.08.014 doi: 10.1016/j.laa.2004.08.014
|
| [3] |
S. Cioabă, D. Gregory, W. Haemers, Matchings in regular graphs from eigenvalues, J. Combin. Theory Ser. B, 99 (2009), 287–297. https://doi.org/10.1016/j.jctb.2008.06.008 doi: 10.1016/j.jctb.2008.06.008
|
| [4] | S. Cioaba, Perfect matchings, eigenvalues and expansion, C. R. Math. Acad. Sci. Soc. R. Can., 27 (2005), 1–6. |
| [5] |
S. Cioabă, D. Gregory, Large matchings from eigenvalues, Linear Algebra Appl., 422 (2007), 308–317. https://doi.org/10.1016/j.laa.2006.10.020 doi: 10.1016/j.laa.2006.10.020
|
| [6] |
O. Suil, Spectral radius and matchings in graphs, Linear Algebra Appl., 614 (2021) 316–324. https://doi.org/10.1016/j.laa.2020.06.004 doi: 10.1016/j.laa.2020.06.004
|
| [7] |
H. Lu, Z. Wu, X. Yang, Eigenvalues and $[1, n]$-odd factors, Linear Algebra Appl., 433 (2010), 750–757. https://doi.org/10.1016/j.laa.2010.04.002 doi: 10.1016/j.laa.2010.04.002
|
| [8] |
S. Kim, S. O, J. Park, H. Ree, An odd $[1, b]$-factor in regular graphs from eigenvalues, Discrete Math., 343 (2020), 111906. https://doi.org/10.1016/j.disc.2020.111906 doi: 10.1016/j.disc.2020.111906
|
| [9] | S. Li, S. Miao, Complete characterization of odd factors via the size, spectral radius or distance spectral radius of graphs, Bull. Korean Math. Soc., 59 (2022), 1045–1067. |
| [10] |
D. Fan, H. Lin, H. Lu, Spectral radius and $[a, b]$-factors in graphs, Discrete Math., 345 (2022), 112892. https://doi.org/10.1016/j.disc.2022.112892 doi: 10.1016/j.disc.2022.112892
|
| [11] |
A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1, b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl., 705 (2025), 1–16. https://doi.org/10.1016/j.laa.2024.10.023 doi: 10.1016/j.laa.2024.10.023
|
| [12] |
D. Fan, H. Lin, Binding number, $k$-factor and spectral radius of graphs, Electron. J. Combin., 31 (2024), 1–26. https://doi.org/10.37236/12165 doi: 10.37236/12165
|
| [13] |
Y. Hao, S. Li, Y. Yu, Bipartite binding number, $k$-factor and spectral radius of bipartite graphs, Discrete Math., 348 (2025), 114511. https://doi.org/10.1016/j.disc.2025.114511 doi: 10.1016/j.disc.2025.114511
|
| [14] |
O. Suil, Eigenvalues and $[a, b]$-factors in regular graphs, J. Graph Theory, 100 (2022), 458–469. https://doi.org/10.1002/jgt.22789 doi: 10.1002/jgt.22789
|
| [15] |
D. Kim, O. Suil, Eigenvalues and parity factors in graphs with given minimum degree, Discrete Math., 346 (2023), 113290. https://doi.org/10.1016/j.disc.2022.113290 doi: 10.1016/j.disc.2022.113290
|
| [16] |
A. Fan, R. Liu, G. Ao, Spectral radius, fractional $[a, b]$-factor and ID-factor-critical graphs, Discrete Math., 347 (2024), 113976. https://doi.org/10.1016/j.disc.2024.113976 doi: 10.1016/j.disc.2024.113976
|
| [17] | E. Cho, J. Hyun, S. O, J. Park, Sharp conditions for the existence of an even $[a, b]$-factor in a graph, Bull. Korean Math. Soc., 58 (2021), 31–46. |
| [18] |
J. Wei, S. Zhang, Proof of a conjecture on the spectral radius condition for $[a, b]$-factors, Discrete Math., 346 (2023), 113269. https://doi.org/10.1016/j.disc.2022.113269 doi: 10.1016/j.disc.2022.113269
|
| [19] |
Y. Hao, S. Li, Turán-type problems on $[a, b]$-factors of graphs, and beyond, Electron. J. Combin., 31 (2024), 1–24. https://doi.org/10.37236/12106 doi: 10.37236/12106
|
| [20] |
D. Fan, R. He, Y. Zhao, Distance spectral radius and edge-disjoint spanning trees, Discrete Appl. Math., 376 (2025), 31–40. https://doi.org/10.1016/j.dam.2025.06.006 doi: 10.1016/j.dam.2025.06.006
|
| [21] |
W. T. Tutte, The factors of graphs, Canad. J. Math., 1 (1952), 314–328. https://doi.org/10.4153/CJM-1952-028-2 doi: 10.4153/CJM-1952-028-2
|
| [22] |
L. Lovász, Subgraphs with prescribed valencies, J. Comb. Theory, 8 (1970), 391–416. https://doi.org/10.1016/S0021-9800(70)80033-3 doi: 10.1016/S0021-9800(70)80033-3
|
| [23] |
J. Folkman, D. R. Fulkerson, Flows in infinite graphs, J. Comb. Theory, 8 (1970), 30–44. https://doi.org/10.1016/S0021-9800(70)80006-0 doi: 10.1016/S0021-9800(70)80006-0
|
Jihan Xin, Huixin Kang, Qi He, Dandan Fan. Distance spectral radius and $ [a, b] $-factor of graphs[J]. AIMS Mathematics, 2026, 11(5): 12360-12372. doi: 10.3934/math.2026507
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