REVIEW 2 major objections 6 minor 49 references
Tur\'an-type problems on $[a,b]$-factors of graphs, and beyond
T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read Graphs with no $[a,b]$-factor have at most $\binom{n-1}{2}+a-1$ edges, and the same extremal graph wins for spectral radius.
desk verdict Genuinely new bipartite results underpin a solid paper; the general-graph theorems lean on an imported lemma and the text has fixable rendering and citation slips. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The proof runs a two-tier degree argument. If $\delta(G)\le a-1$, the graph embeds in $K_{a-1}\vee(K_{n-a}\cup K_1)$, which immediately yields the edge bound; if $\delta(G)\ge a$, an imported threshold lemma [44] says that $e(G)\ge \binom{n-1}{2}+\frac{a+1}{2}$ forces an $[a,b]$-factor when $na$ is even in the case $a=b$, so a factor-free graph must fall strictly below this threshold. The bipartite arguments use the $(g,f)$-factor criterion of [17]: a bipartite graph has an $[a,b]$-factor if and only if $b|S|+\sum_{v\in T}d_G(v)-a|T|-e(S,T)\ge 0$ and its mirror inequality hold for all subsets $S\subseteq X$ and $T\subseteq Y$; violating this criterion supplies the inequalities that bound the edge count. The extremal bipartite shapes are double nested graphs, in which the $i$-th block of one part is joined to the first several blocks of the other part, and their spectral radii are compared through equitable quotient matrices and the associated characteristic polynomials.
What would settle it
Run an exhaustive search at the smallest nontrivial parameters, e.g., $a=1$, $b=1$, $n=6$: the imported lemma predicts that every 6-vertex graph with $\delta(G)\ge 1$ and at least $\binom{5}{2}+1=11$ edges contains a perfect matching, so any such graph without one would refute the engine behind Theorems 1 and 2. For the bipartite results, check whether any graph with part sizes $p\le q$, $a\le p$, $aq\le bp$, and $e(G)=p(q-1)+a$ fails to have an $[a,b]$-factor; Theorem 3 predicts none, and a single violation would break the $(g,f)$-criterion argument used in the bipartite proofs.
Extended reading notes
Core claim
The central discovery is a sharp phase transition: an $n$-vertex graph contains an $[a,b]$-factor as soon as its edge count reaches $\binom{n-1}{2}+a$, provided $n\ge a+1$ and, when $a=b$, $na$ is even. The unique obstruction below that threshold is the join $K_{a-1}\vee(K_{n-a}\cup K_1)$, consisting of a clique on $a-1$ universal vertices joined to a clique on $n-a$ vertices plus one isolated vertex; the exceptional graphs $K_{1,3}$ and $K_2\vee K_3$ cover the cases $ab\le 2$ and $a=b=2$ respectively. The same join is the unique spectral extremal graph, so a graph whose spectral radius reaches $\rho(K_{a-1}\vee(K_{n-a}\cup K_1))$ must contain an $[a,b]$-factor unless it is exactly that graph. In the bipartite setting with parts of sizes $p\le q$, the extremal graph is the complete bipartite graph $K_{p,q}$ when $aq>bp$ or $a>p$, and otherwise the double nested graph $D(a-1,p-a+1;q-1,1)$; for a fixed total number of vertices the answer is whichever of the corresponding complete bipartite graph and nested graph has more edges or larger spectral radius.
Load-bearing premise
The load-bearing premise is the imported lemma [44] asserting that every $n$-vertex graph with $\delta(G)\ge a$ and $e(G)\ge \binom{n-1}{2}+\frac{a+1}{2}$ contains an $[a,b]$-factor, with $na$ even when $a=b$; the paper does not reprove this lemma, so a failure in any degree or parity range would require revising Theorems 1 and 2.
Editorial extensions
If this is right
- Every $n$-vertex graph with more than $\binom{n-1}{2}+a-1$ edges contains an $[a,b]$-factor, and the graphs listed in Theorem 1 are the only factor-free graphs attaining the bound.
- Every $n$-vertex graph whose spectral radius is at least $\rho(K_{a-1}\vee(K_{n-a}\cup K_1))$ contains an $[a,b]$-factor unless it is exactly that join, giving a spectral analogue of the edge threshold.
- Directly, $\mathrm{Ex}_{sp}(n,\mathcal{F}_{a,b})\subseteq \mathrm{Ex}(n,\mathcal{F}_{a,b})$ for all admissible $a,b,n$, and the same containment holds for bipartite graphs, resolving the containment problem of [29] for these families.
- In the balanced bipartite case $a=b=k$, the spectral bound recovers the $k$-factor theorem of [15].
Reading between the lines
- The same equitable-quotient technique used for $D(a-1,p-a+1;q-1,1)$ should transfer to other bipartite spectral extremal problems whose candidates are a complete bipartite graph and a nested chain graph; the deciding comparison is a single polynomial inequality.
- The bipartite proof is driven entirely by the $(g,f)$-factor criterion, so the results should extend to $(g,f)$-factors with non-constant degree intervals by replacing the constants $a,b$ with vertex-dependent functions in the extremal inequalities.
- With $\delta(G)\ge a$ imposed, the extremal join has an isolated vertex and cannot be extremal; determining the true extremal graphs for that minimum-degree variant, posed as Problem 2 in the paper, would require new arguments and could be explored computationally for small $a,b$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies Turán-type problems for graphs with no [a,b]-factor, i.e., no spanning subgraph whose degrees all lie in the interval [a,b]. It determines, for n-vertex graphs, the maximum number of edges ex(n,F_{a,b}) and the maximum adjacency spectral radius ex_sp(n,F_{a,b}), together with the complete lists of extremal graphs, under the parity condition na≡0 (mod 2) when a=b. It also obtains the bipartite analogues, both with fixed partite sizes and with fixed total order, again with full extremal characterizations. As a byproduct, it shows that the spectral extremal graphs lie inside the edge-extremal graphs, contributing to a problem of Liu and Ning, and it deduces a main result of Fan and Lin on spectral conditions for k-factors in balanced bipartite graphs. The proofs use the Folkman-Fulkerson (g,f)-factor criterion, Ore's Hamiltonian theorems, Rowlinson's spectral bound, Liu-Weng's bound for dense bipartite graphs, and detailed case analyses.
Significance. If correct, these results completely settle the edge and spectral extremal problems for the family of [a,b]-factors, a natural and broad class of spanning subgraphs. The explicit identification of all extremal graphs, including the bipartite double nested graphs, is valuable, and the inclusion Ex_sp⊆Ex is a nontrivial positive contribution to Problem 1 of Liu and Ning. The paper also unifies and extends earlier results on k-factors and [a,b]-factors. The proofs are largely self-contained apart from the imported Lemma 8, and the case analyses in the bipartite proofs are detailed and checkable. The spectral comparisons via equitable partitions and characteristic polynomials are concrete and avoid black-box extremal arguments.
major comments (2)
- [Theorem 1(ii) and Lemma 10] As printed, Theorem 1(ii) and Lemma 10 state the exceptional graph as K2∨K3. Since K2∨K3 is the complete graph K5, it contains a Hamiltonian cycle and hence a [2,2]-factor; it therefore cannot be an extremal graph for [2,2]-factor-free graphs. The intended graph is evidently K2∨\overline{K3}, the join of K2 with the complement of K3. Please correct this typographical error in Theorem 1(ii), Lemma 10, and any subsequent references (e.g., the discussion in Section 6). This is not merely cosmetic: as written, Theorem 1 is false for n=5, a=b=2.
- [Section 3, Lemma 8] The proof of Theorem 1 in the case δ(G)≥a rests entirely on Lemma 8, which is quoted from [44] without proof and without a precise reference (theorem number or page). The exact threshold binom(n-1,2)+(a+1)/2 and the parity condition na≡0 (mod 2) when a=b are load-bearing: the contrapositive gives e(G)≤binom(n-1,2)+a/2, which in turn yields the claimed bound binom(n-1,2)+a-1 for a≥3. If the lemma as quoted from [44] has any additional hidden hypothesis (e.g., a lower bound on n, or a connectedness assumption), Theorems 1 and 2 would require revision. Please either provide a proof of Lemma 8 in the paper or give the exact statement and theorem number in [44], and confirm that the statement is correctly reproduced.
minor comments (6)
- [Section 5, equations (5.2) and (5.3)] In the displayed characteristic polynomials Φ2(x) and Φ3(x), the symbol "t2" should be "x^2"; the current rendering makes the algebra difficult to follow.
- [Section 4, proof of Theorem 3, Case 1] When u∈X, the graph D(p−1,1; a−1,q−a+1) is isomorphic to D(a−1,p−a+1;q−1,1) by interchanging the bipartition and reversing the order of the parts; stating this explicitly would remove the apparent mismatch with the theorem's equality case.
- [Section 5, proof of Theorem 5(iii)] The sentence "by Theorem 3(iii), G contains a [1,b]-factor" is a non-sequitur in isolation; the intended argument is the contrapositive: if G had no [1,b]-factor, then equality in Theorem 3(iii) would force G≅G1, contradicting the assumption. Please rephrase this step.
- [Theorems 1 and 4] The exceptional graphs K1,3 and K2∨\overline{K3} have orders 4 and 5 respectively; a remark noting that these graphs can appear as equality cases only for those orders would prevent the reader from thinking they occur for all n.
- [Section 4, proof of Theorem 4, Case 2] The equality analysis concludes that G≅K_{n/2−1,n/2+1}; it would help to note that this graph coincides with D(a−1,n/2−a;n/2,1) when a=n/2, which is already in the list of extremal graphs.
- [Section 5, proof of Theorem 7] The proof uses "by a direct computation" for the characteristic polynomials in (5.1)–(5.3); since these computations are central to the spectral comparison, adding the intermediate simplification steps would improve verifiability.
Circularity Check
No significant circularity: the main theorems are derived from external factor criteria and standard spectral lemmas, and the target results are not assumed as inputs.
full rationale
The derivation chain is self-contained relative to its cited external tools. Theorem 1 splits into the low-degree case δ(G) ≤ a−1, which is an elementary spanning-subgraph bound, and the high-degree case δ(G) ≥ a, which invokes Lemma 8 from Wei and Zhang [44]. Lemma 8 is an external result, not a theorem proved or assumed by the present authors, and its contrapositive is applied to convert the absence of an [a,b]-factor into an edge bound; this is a legitimate dependency, not a circular reduction. The equality cases are then handled by Ore's Hamilton-path and Hamilton-cycle theorems (Lemmas 9 and 10), again external. Theorem 2 reduces the spectral problem to Theorem 1 through Hong's bound (Lemma 3) and Rowlinson's extremal spectral result (Lemma 4); no fitted parameter or normalization forces the conclusion. The bipartite Theorems 3–7 are built on the Folkman–Fulkerson (g,f)-factor criterion (Theorem 8/Corollary 4) and on Liu–Weng's spectral bound (Lemma 5), with all remaining case analysis performed in the paper. The statement that Theorem 2 strengthens a result of Wei and Zhang is not circular because the proof does not use the same spectral theorem as its premise. Self-citations in the discussion section (e.g., [24], [25], [31]) support illustrative observations and a conjecture, but they are not load-bearing for the main theorems. The most serious verification concern is the unproved importation of Lemma 8 with its sharp threshold and parity condition, but a dependence on an external lemma is a correctness risk, not a circularity. Consequently, no step in the paper reduces the claimed predictions to their own inputs, and no existing result is merely renamed as a new theorem.
Assumptions & free parameters
assumptions (5)
- domain assumption Lemma 8 (Wei-Zhang): if delta(G) >= a and e(G) >= (n-1 choose 2) + (a+1)/2 with na even when a=b, then G has an [a,b]-factor.
- standard math Folkman-Fulkerson theorem: a bipartite graph has a (g,f)-factor iff both inequalities over S subset X and T subset Y hold.
- standard math Ore's Hamilton path and Hamilton cycle theorems with their exceptional graphs.
- standard math Rowlinson's theorem: among graphs with n vertices and (r choose 2)+t edges, the unique spectral extremal graph is K_t vee (K_{r-t} union K_1) plus isolated vertices.
- standard math Hong's bound rho(G) <= sqrt(2e(G)-n+1), and Liu-Weng's spectral bound for bipartite graphs with pq-p < e < pq.
Cite this review
Pith. "Pith review of Tur\'an-type problems on $[a,b]$-factors of graphs, and beyond." pith.science (2026). https://pith.science/paper/ZIKDGWKE
@misc{pith2026241116143,
author = {Pith},
title = {Pith review of: Tur\'an-type problems on $[a,b]$-factors of graphs, and beyond},
year = {2026},
howpublished = {\url{https://pith.science/paper/ZIKDGWKE}},
note = {Machine review of arXiv:2411.16143}
}
abstract
Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the maximum size (resp. spectral radius) of an $n$-vertex $\mathcal{H}$-free graph. Denote by $\text{Ex}(n, \mathcal{H})$ the set of all $n$-vertex $\mathcal{H}$-free graphs with $\text{ex}(n, \mathcal{H})$ edges. Similarly, let $\mathrm{Ex}_{sp}(n,\mathcal{H})$ be the set of all $n$-vertex $\mathcal{H}$-free graphs with spectral radius $\text{ex}_{sp}(n, \mathcal{H})$. For positive integers $a, b$ with $a\leqslant b$, an $[a,b]$-factor of a graph $G$ is a spanning subgraph $F$ of $G$ such that $a\leqslant d_F(v)\leqslant b$ for all $v\in V(G)$, where $d_F(v)$ denotes the degree of the vertex $v$ in $F.$ Let $\mathcal{F}_{a,b}$ be the set of all the $[a,b]$-factors of an $n$-vertex complete graph $K_n$. In this paper, we determine the Tur\'an number $\text{ex}(n,\mathcal{F}_{a,b})$ and the spectral Tur\'an number $\text{ex}_{sp}(n,\mathcal{F}_{a,b}),$ respectively. Furthermore, the bipartite analogue of $\text{ex}(n,\mathcal{F}_{a,b})$ (resp. $\text{ex}_{sp}(n,\mathcal{F}_{a,b})$) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that $\mathrm{Ex}_{sp}(n,\mathcal{F}_{a,b})\subseteq \text{Ex}(n, \mathcal{F}_{a,b})$ holds for graphs and bipartite graphs. This partially answers an open problem proposed by Liu and Ning \cite{LN2023}. Our results may deduce a main result of Fan and Lin \cite{FL2022}.
Figures
Reference graph
Works this paper leans on
-
[34]
O, Spectral radius and matchings in graphs, Linear Algebra A ppl
S. O, Spectral radius and matchings in graphs, Linear Algebra A ppl. 614 (2020) 316-324. 21
work page 2020
- [44]
-
[1]
N. Alon, R. Yuster, The Tur´ an number of sparse spanning grap hs, J. Combin. Theory Ser. B 103 (3) (2013) 337-343
work page 2013
-
[2]
M. Andeli´ c, C.M. Da Fonseca, T. Koledin, Z. Stani´ c, Sharp spectral inequalities for connected bipartite graphs with maximal Q-index, Ars Math. Contemp. 6 (2013) 171-185
work page 2013
-
[3]
Bapat, Graphs and Matrices, Springer, New York, 2010
R.B. Bapat, Graphs and Matrices, Springer, New York, 2010
2010
-
[4]
A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalu e of bipartite graphs, Electron. J. Combin. 15 (1) (2008) 144. 20
work page 2008
-
[5]
Brouwer, W.H
A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, New York, 2012
2012
-
[6]
J. Byrne, D.N. Desai, M. Tait, A general theorem in spectral ext remal graph theory, arXiv:2401.07266
Show all 49 references
-
[7]
Chen, R.J
G.T. Chen, R.J. Gould, F. Pfender, B. Wei, Extremal graphs for in tersecting cliques, J. Combin. Theory Ser. B 89 (2) (2003) 159-171
2003
-
[8]
Cho, J.Y
E.-K. Cho, J.Y. Hyun, S. O, J.R. Park, Sharp conditions for the ex istence of an even [ a, b ]-factor in a graph, Bull. Korean Math. Soc. 58 (1) (2021) 31-46
2021
-
[9]
Cioab˘ a, D.N
S. Cioab˘ a, D.N. Desai, M. Tait, The spectral radius of graphs wit h no odd wheels, European J. Combin. 99 (2022) 103420
2022
-
[10]
Cioab˘ a, L.H
S. Cioab˘ a, L.H. Feng, M. Tait, X.-D. Zhang, The maximum spectr al radius of graphs without friendship subgraphs, Electron. J. Combin. 27 (4) (2020) #P4.22
2020
-
[11]
Desai, L.Y
D.N. Desai, L.Y. Kang, Y.T. Li, Z.Y. Ni, M. Tait, J. Wang, Spectral e xtremal graphs for intersecting cliques, Linear Algebra Appl. 644 (2022) 234-258
2022
-
[12]
Erd˝ os, Z
P. Erd˝ os, Z. F¨ uredi, R.J. Gould, D. S. Gunderson, Extremalgraphs for intersecting triangles, J. Combin. Theory Ser. B 64 (1995) 89-100
1995
-
[13]
Erd˝ os, M
P. Erd˝ os, M. Simonovits, A limit theorem in graph theory, Studia Sci. Math. Hungar. 1 (1966) 51-57
1966
-
[14]
Erd˝ os, A.H
P. Erd˝ os, A.H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946) 1087-1091
1946
-
[15]
Fan, H.Q
D.D. Fan, H.Q. Lin, Spectral conditions for k-extendability and k-factors of bipartite graphs, arXiv:2211.09304v1 [math. CO]. Available at https://arxiv.org/pdf/2211.09304.pdf
-
[16]
Fiedler, V
M. Fiedler, V. Nikiforov, Spectral radius and Hamiltonicity of gra phs, Linear Algebra Appl. 432 (9) (2010) 2170-2173
2010
-
[17]
Folkman, D.R
J. Folkman, D.R. Fulkerson, Flows in infinite graphs, J. Combinato rial Theory 8 (1970) 30-44
1970
-
[18]
F¨ uredi, M
Z. F¨ uredi, M. Simonovits, The history of degenerate (bipartit e) extremal graph problems, Erd˝ os cen- tennial, Bolyai Soc. Math. Stud., 25 (2013) 169-264
2013
-
[19]
Godsil, G
C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Texts in M athematics, vol. 207, Springer-Verlag, New York, 2001
2001
-
[20]
Guiduli, Spectral Extrema for Graphs, Ph.D
B. Guiduli, Spectral Extrema for Graphs, Ph.D. Thesis, Depart ment of Mathematics, University of Chicago, 1996. Available at http://people.cs.uchicago.edu/~laci/students
1996
-
[21]
Hao, S.C
Y.F. Hao, S.C. Li, X.C. Li, Vertex cut, eigenvalues, [ a, b ]-factors and toughness of connected bipartite graphs, Discrete Math. 347 (10) (2024) 114118
2024
-
[22]
Hong, A bound on the spectral radius of graphs, Linear Alge bra Appl
Y. Hong, A bound on the spectral radius of graphs, Linear Alge bra Appl. 108 (1988) 135-139
1988
-
[23]
Lei, S.C
X.Y. Lei, S.C. Li, Spectral extremal problem on disjoint color-cr itical graphs, Electron. J. Combin. 31 (1) (2024) #P1.25
2024
-
[24]
S.C. Li, S.J. Miao, Characterizing P/greaterorequalslant2-factor and P/greaterorequalslant2-factor covered graphs with respect to the size or the spectral radius, Discrete Math. 344 (11) (2021) Paper No. 1 12588
2021
-
[25]
S.C. Li, S.J. Miao, Complete characterization of odd factors via t he size, spectral radius or distance spectral radius of graphs, Bull. Korean Math. Soc. 59 (4) (2022) 1045-1067
2022
-
[26]
S.C. Li, W.T. Sun, Some spectral inequalities for connected bipar tite graphs with maximum Aα -index, Discrete Appl. Math. 287 (2020) 97-109
2020
-
[27]
Y.T. Li, L.H. Feng, W.J. Liu, A survey on spectral conditions for s ome extremal graph problem, Adv. Math. (China) 51 (2) (2022) 193-258
2022
-
[28]
Liu, C.-W
C.-A. Liu, C.-W. Weng, Spectral radius of bipartite graphs, Line ar Algebra Appl. 474 (2015) 30-43
2015
-
[29]
L.L. Liu, B. Ning, Spectral Tur´ an-type problems on sparse sp anning graphs, arXiv:2307.14629
-
[30]
Mantel, Problem 28, Wiskundige Opgaven, 10 (60-61) (1907) 320
W. Mantel, Problem 28, Wiskundige Opgaven, 10 (60-61) (1907) 320
1907
-
[31]
Miao, S.C
S.J. Miao, S.C. Li, Characterizing star factors via the size, the s pectral radius or the distance spectral radius of graphs, Discrete Appl. Math. 326 (2023) 17-32
2023
-
[32]
Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Ap pl
V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Ap pl. 427 (2007) 183-189
2007
-
[33]
Nikiforov, Some new results in extremal graph theory
V. Nikiforov, Some new results in extremal graph theory. Surv eys in Combinatorics 2011, 141-181, Cambridge University Press, Cambridge, 2011
2011
-
[35]
Ore, Arc coverings of graphs, Ann
O. Ore, Arc coverings of graphs, Ann. Mat. Pura Appl. 55 (4) ( 1961) 315-321
1961
-
[36]
Petrovi´ c, S.K
M. Petrovi´ c, S.K. Simi´ c, A note on connected bipartite grap hs of fixed order and size with maximal index, Linear Algebra Appl. 483 (2015) 21-29
2015
-
[37]
Rowlinson, On the maximal index of graphs with a prescribed nu mber of edges, Linear Algebra Appl
P. Rowlinson, On the maximal index of graphs with a prescribed nu mber of edges, Linear Algebra Appl. 110 (1988) 43-53
1988
-
[38]
Simonovits, Extremal graph problems with symmetrical extr emal graphs, Additional chromatic conditions, Discrete Math
M. Simonovits, Extremal graph problems with symmetrical extr emal graphs, Additional chromatic conditions, Discrete Math. 7 (1974) 349-376
1974
-
[39]
Sun, S.C
W.T. Sun, S.C. Li, W. Wei, Extensions on spectral extrema of C5/C 6-free graphs with given size, Discrete Math. 346 (12) (2023) 113591
2023
-
[40]
Sun, S.C
W.T. Sun, S.C. Li, W. Wei, Forbidden theta graph, bounded spect ral radius and size of non-bipartite graphs, J. Korean Math. Soc. 60 (5) (2023) 959-986
2023
-
[41]
Tur´ an, On an extremal problem in graph theory, Mat
P. Tur´ an, On an extremal problem in graph theory, Mat. Fiz. L apok, 48 (1941) 436-452
1941
-
[42]
Wang, L.Y
J. Wang, L.Y. Kang, Y.S. Xue, On a conjecture of spectral ext remal problems, J. Combin. Theory Ser. B 159 (2023) 20-41
2023
-
[43]
Wang, Z.Y
J. Wang, Z.Y. Ni, L.Y. Kang, Y.Z. Fan, Spectral extremal graph s for edge blow-up of star forests, Discrete Math. 347 (10) (2024) 114141
2024
-
[45]
West, Introduction to Graph Theory, Prentice Hall, Inc., U pper Saddle River, NJ, 1996
D.B. West, Introduction to Graph Theory, Prentice Hall, Inc., U pper Saddle River, NJ, 1996
1996
-
[46]
D.L. You, J. Wang, L.Y. Kang, Spectral extremal graph for ed ge blow-up of star forests (Chinese), submitted
-
[47]
L.H. You, M. Yang, W. So, W.G. Xi, On the spectrum of an equitable quotient matrix and its application, Linear Algebra Appl. 577 (2019) 21-40
2019
-
[48]
Zhai, H.Q
M.Q. Zhai, H.Q. Lin, A strengthening of the spectral chromatic c ritical edge theorem: Books and theta graphs, J. Graph Theory 102 (3) (2023) 502-520
2023
-
[49]
Zhai, R.F
M.Q. Zhai, R.F. Liu, J. Xue, A unique characterization of spectra l extrema for friendship graphs, Electron. J. Combin. 29 (3) (2022) #P3.32. 22
2022
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.