Graphs whose spectral radius is bounded by a fixed Hoffman–Smith limit point
For an integer p ≥ 3 , we study graphs whose adjacency spectral radius satisfies ρ ( G ) < p p - 1 = A p or whose signless Laplacian spectral radius satisfies κ ( G ) < p 2 p - 1 = Q p . The numbers A p and Q p are known as Hoffman–Smith limit points. For general p , we find upper bounds on th...
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| Veröffentlicht in: | Journal of algebraic combinatorics Jg. 62; H. 2; S. 25 |
|---|---|
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| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
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Springer US
01.09.2025
Springer Nature B.V |
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| ISSN: | 0925-9899, 1572-9192 |
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| Abstract | For an integer
p
≥
3
, we study graphs whose adjacency spectral radius satisfies
ρ
(
G
)
<
p
p
-
1
=
A
p
or whose signless Laplacian spectral radius satisfies
κ
(
G
)
<
p
2
p
-
1
=
Q
p
. The numbers
A
p
and
Q
p
are known as Hoffman–Smith limit points. For general
p
, we find upper bounds on the maximum degree of such graphs
G
and describe the graphs for which the upper bound is achieved with respect to the adjacency matrix. Moreover, we describe forbidden substructures for graphs in these classes. For
p
=
4
, we show that the structure of graphs
G
such that
A
3
<
ρ
(
G
)
<
A
4
or
Q
3
<
κ
(
G
)
<
Q
4
is much richer than the structure of graphs for which
ρ
(
G
)
<
A
3
or
κ
(
G
)
<
Q
3
, whose study has been initiated by Woo and Neumaier (Graphs Combin 23:713–726, 2007). |
|---|---|
| AbstractList | For an integer p≥3, we study graphs whose adjacency spectral radius satisfies ρ(G)<pp-1=Ap or whose signless Laplacian spectral radius satisfies κ(G)<p2p-1=Qp. The numbers Ap and Qp are known as Hoffman–Smith limit points. For general p, we find upper bounds on the maximum degree of such graphs G and describe the graphs for which the upper bound is achieved with respect to the adjacency matrix. Moreover, we describe forbidden substructures for graphs in these classes. For p=4, we show that the structure of graphs G such that A3<ρ(G)<A4 or Q3<κ(G)<Q4 is much richer than the structure of graphs for which ρ(G)<A3 or κ(G)<Q3, whose study has been initiated by Woo and Neumaier (Graphs Combin 23:713–726, 2007). For an integer p ≥ 3 , we study graphs whose adjacency spectral radius satisfies ρ ( G ) < p p - 1 = A p or whose signless Laplacian spectral radius satisfies κ ( G ) < p 2 p - 1 = Q p . The numbers A p and Q p are known as Hoffman–Smith limit points. For general p , we find upper bounds on the maximum degree of such graphs G and describe the graphs for which the upper bound is achieved with respect to the adjacency matrix. Moreover, we describe forbidden substructures for graphs in these classes. For p = 4 , we show that the structure of graphs G such that A 3 < ρ ( G ) < A 4 or Q 3 < κ ( G ) < Q 4 is much richer than the structure of graphs for which ρ ( G ) < A 3 or κ ( G ) < Q 3 , whose study has been initiated by Woo and Neumaier (Graphs Combin 23:713–726, 2007). |
| ArticleNumber | 25 |
| Author | Calegari, Rafael Hoppen, Carlos Trevisan, Vilmar Borba, Elizandro Max Veloso, Bruno Scaratti |
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| Cites_doi | 10.1016/j.laa.2025.01.044 10.1017/CBO9781139086547 10.1016/j.laa.2010.08.006 10.1007/s00373-007-0745-9 10.1016/j.laa.2011.05.006 10.13001/ela.2024.7979 10.1007/s10801-021-01090-2 10.1007/BFb0066438 10.1016/j.laa.2010.05.010 10.1080/10236198.2021.1962315 10.1016/j.laa.2009.09.016 10.2298/PIM0999019C 10.1016/j.laa.2009.02.017 10.1002/jgt.21690 10.1007/978-3-031-11698-8 10.1016/j.aam.2025.102915 |
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| Keywords | Hoffman–Smith limit points Hoffman program Graph subdivision |
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| References | SM Cioabă (1458_CR6) 2010; 432 1458_CR15 A Brouwer (1458_CR4) 2011 F Belardo (1458_CR1) 2022; 55 AJ Hoffman (1458_CR10) 1975 DP Jacobs (1458_CR13) 2011; 434 H Kumar (1458_CR14) 2025; 710 D Cvetković (1458_CR7) 1997 C Hoppen (1458_CR12) 2024; 40 ER Oliveira (1458_CR16) 2021; 27 C Godsil (1458_CR9) 2013 Y Chen (1458_CR5) 2013; 74 F Belardo (1458_CR3) 2011; 435 J Wang (1458_CR18) 2009; 431 J Wang (1458_CR19) 2025; 169 F Belardo (1458_CR2) 2010; 433 C Hoppen (1458_CR11) 2022 AJ Schwenk (1458_CR17) 1974 R Woo (1458_CR20) 2007; 23 D Cvetković (1458_CR8) 2009; 85 |
| References_xml | – volume: 710 start-page: 336 year: 2025 ident: 1458_CR14 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2025.01.044 – volume-title: Eigenspaces of Graphs year: 1997 ident: 1458_CR7 doi: 10.1017/CBO9781139086547 – volume: 434 start-page: 81 issue: 1 year: 2011 ident: 1458_CR13 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2010.08.006 – volume: 23 start-page: 713 year: 2007 ident: 1458_CR20 publication-title: Graphs Combin. doi: 10.1007/s00373-007-0745-9 – volume: 435 start-page: 2913 issue: 11 year: 2011 ident: 1458_CR3 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2011.05.006 – volume: 40 start-page: 81 year: 2024 ident: 1458_CR12 publication-title: Electron. J. Linear Algebra doi: 10.13001/ela.2024.7979 – volume: 55 start-page: 1199 issue: 4 year: 2022 ident: 1458_CR1 publication-title: J. Algebraic Combin. doi: 10.1007/s10801-021-01090-2 – start-page: 153 volume-title: Graphs and Combinatorics year: 1974 ident: 1458_CR17 doi: 10.1007/BFb0066438 – volume: 433 start-page: 1513 issue: 11 year: 2010 ident: 1458_CR2 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2010.05.010 – volume: 27 start-page: 1024 issue: 7 year: 2021 ident: 1458_CR16 publication-title: J. Difference Equ. Appl. doi: 10.1080/10236198.2021.1962315 – volume: 432 start-page: 722 issue: 2 year: 2010 ident: 1458_CR6 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2009.09.016 – volume-title: Algebraic Graph Theory year: 2013 ident: 1458_CR9 – volume-title: Spectra of Graphs year: 2011 ident: 1458_CR4 – ident: 1458_CR15 doi: 10.1080/10236198.2021.1962315 – volume: 85 start-page: 19 issue: 99 year: 2009 ident: 1458_CR8 publication-title: Publ. Inst. Math. (Beograd) (N.S.) doi: 10.2298/PIM0999019C – start-page: 273 volume-title: Recent Advanced in Graph Theory (Prague, 1975) year: 1975 ident: 1458_CR10 – volume: 431 start-page: 162 issue: 1 year: 2009 ident: 1458_CR18 publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2009.02.017 – volume: 74 start-page: 32 issue: 1 year: 2013 ident: 1458_CR5 publication-title: J. Graph Theory doi: 10.1002/jgt.21690 – volume-title: Locating Eigenvalues in Graphs: Algorithms and Applications year: 2022 ident: 1458_CR11 doi: 10.1007/978-3-031-11698-8 – volume: 169 year: 2025 ident: 1458_CR19 publication-title: Adv. Appl. Math. doi: 10.1016/j.aam.2025.102915 |
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| Snippet | For an integer
p
≥
3
, we study graphs whose adjacency spectral radius satisfies
ρ
(
G
)
<
p
p
-
1
=
A
p
or whose signless Laplacian spectral radius satisfies... For an integer p≥3, we study graphs whose adjacency spectral radius satisfies ρ(G)<pp-1=Ap or whose signless Laplacian spectral radius satisfies κ(G)<p2p-1=Qp.... |
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| SubjectTerms | Combinatorics Computer Science Convex and Discrete Geometry Eigenvalues Graphs Group Theory and Generalizations Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures Upper bounds |
| Title | Graphs whose spectral radius is bounded by a fixed Hoffman–Smith limit point |
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