Dense-core approach to the Brualdi--Hoffman--Turán problem on odd wheels

Abstract: We present a unified presentation of the fixed-size adjacency-spectral extremal problem for odd wheels $W_{2k+1}$, where $k\geq2$ and $W_{2k+1}=K_1\vee C_{2k}$. The exceptional case $W_5$ and the general case $W_{2k+1}$, $k\ge3$, share the same dense-core reduction and edge-spectral stability, but have different rigidity structures. We prove that every $W_5$-free graph of sufficiently large size $m$ satisfies $ρ(G)^2-ρ(G)\le m,$ with equality precisely for $K_{n,n}$ with a perfect matching embedded in each part, where $n$ is even and $m=n<sup>2+n$. For any fixed $k\ge3$, every $W_{2k+1}$-free graph of sufficiently large size $m$ satisfies $ρ(G)^{2-(k-1)ρ(G)\le} m-\binom{k}{2},$ with equality precisely for $K_k\vee qK_1$ when $m=\binom{k}{2}+kq$. Our results completely settle a conjecture proposed by Yu, Li and Peng and, via a distinct approach, further strengthen known results concerning odd cycles, friendship graphs and odd fan graphs for sufficiently large $m.$ The proof combines the edge-spectral stability theorem, residual functions and the dense-core method.

• The paper resolves the final Yu–Li–Peng conjecture, proving sharp asymptotic spectral bounds for graphs avoiding odd wheels and characterizing all equality cases.
• For W₅, the extremal graph is a balanced complete bipartite graph with a perfect matching added inside each part, while for k≥3 it is the split graph Kₖ∨qK₁.
• The authors combine edge-spectral stability, dense-core reduction, Perron localization, and residual identities to show that forbidden even cycles in vertex neighborhoods force these rigid extremal structures.

The Brualdi–Hoffman–Turán problem for odd wheels

The Brualdi–Hoffman–Turán problem asks for the maximum spectral radius of an $m$-edge graph subject to a Turán-type subgraph constraint. This paper by Fang, Zhai, and Zhang resolves the last open member of a family of conjectures posed by Yu, Li, and Peng: the fixed-size spectral extremal problem for odd wheels $W_{2k+1}=K_1\vee C_{2k}$. The two companion conjectures concerning fan graphs $V_{2k+1}$ and even wheels $W_{2k+2}$ had already been settled — the former by Li, Zhao, and Zou via the extremal graph $K_k\vee qK_1$, the latter by Li, Liu, and Zhang through their edge-spectral Erdős–Stone–Simonovits theorem for color-critical graphs. The odd-wheel case resisted these techniques because forbidding a single odd wheel is a strictly weaker condition than forbidding a fan or an even wheel: the neighborhood of every vertex must merely be $C_{2k}$-free, which permits a nonlinear number of edges locally.

The paper establishes two main theorems with exact equality characterizations. For the exceptional wheel $W_5$, there exists $m_0$ such that every $W_5$-free graph $G$ with $W_{2k+1}=K_1\vee C_{2k}$0 edges satisfies $W_{2k+1}=K_1\vee C_{2k}$1, with equality if and only if $W_{2k+1}=K_1\vee C_{2k}$2 is obtained from $W_{2k+1}=K_1\vee C_{2k}$3 ($W_{2k+1}=K_1\vee C_{2k}$4 even, $W_{2k+1}=K_1\vee C_{2k}$5) by embedding a perfect matching inside each part. For each fixed $W_{2k+1}=K_1\vee C_{2k}$6, every $W_{2k+1}=K_1\vee C_{2k}$7-free graph of sufficiently large size satisfies $W_{2k+1}=K_1\vee C_{2k}$8, with equality precisely at the split graph $W_{2k+1}=K_1\vee C_{2k}$9 where $V_{2k+1}$0. These bounds are attained by the functions $V_{2k+1}$1 and $V_{2k+1}$2 respectively, confirming Conjecture 2 of Yu–Li–Peng in full.

Sharpness and the structural dichotomy

A notable feature of the result is that the extremal configurations differ qualitatively between $V_{2k+1}$3 and $V_{2k+1}$4. For $V_{2k+1}$5, the balanced bipartite construction $V_{2k+1}$6 plus internal perfect matchings is $V_{2k+1}$7-regular, hence has spectral radius exactly $V_{2k+1}$8; it remains $V_{2k+1}$9-free because the neighborhood of any vertex is a join of its matching partner with a matching in the opposite part, which contains no $W_{2k+2}$0. For $W_{2k+2}$1, the local obstruction strengthens: the longest cycle in $W_{2k+2}$2 for the split extremal graph has length below $W_{2k+2}$3, so $W_{2k+2}$4 is $W_{2k+2}$5-free while attaining the threshold identity $W_{2k+2}$6 directly from the eigen-equations on the clique and independent parts.

The authors also prove a standalone rigidity lemma for the $W_{2k+2}$7 case: any graph whose vertex set partitions into two sets inducing maximum degree at most one satisfies $W_{2k+2}$8, with equality forcing the regular matched complete bipartite structure. The equality analysis proceeds by multiplying paired Perron-coordinate inequalities across the bipartition and tracing saturation conditions to force constant Perron coordinates, hence regularity, then degree counting to force $W_{2k+2}$9 with all cross edges present.

Methodology

Three tools are combined. First, the edge-spectral stability theorem of Li–Liu–Zhang: for fixed $K_k\vee qK_1$0 with $K_k\vee qK_1$1, any $K_k\vee qK_1$2-free graph with $K_k\vee qK_1$3 large edges and $K_k\vee qK_1$4 is within $K_k\vee qK_1$5 edge edits of a complete bipartite graph $K_k\vee qK_1$6, and moreover $K_k\vee qK_1$7 for all such graphs. Second, the dense-core method of Fang–Lin–Zhai: iteratively deleting $K_k\vee qK_1$8-dense proper subgraphs (those increasing the ratio $K_k\vee qK_1$9 by at least the proportional edge loss) terminates at an $C_{2k}$0-core retaining $C_{2k}$1 edges and spectral radius above the target function. A key technical contribution here is Lemma 3's reduction argument, which shows the core's spectral ratio exceeds $C_{2k}$2 by comparing the accumulated gain $C_{2k}$3 against the drift of $C_{2k}$4, controlled via $C_{2k}$5. Third, the residual function of Zhai–Li–Lou, which converts the eigen-equation at a Perron-maximizing vertex into an identity isolating the surplus $C_{2k}$6 as a sum of signed local terms.

On top of this common machinery sits a detailed Perron localization analysis. The minimum-edit partition is refined so that $C_{2k}$7 is shown empty by a contradiction between a lower bound $C_{2k}$8 and upper bounds derived from the subset estimates $C_{2k}$9. The analysis then splits into a split regime ($W_5$0), where the Perron mass concentrates in the smaller part and vertices of $W_5$1 carry coordinates at most a quarter of the maximum, and a balanced regime ($W_5$2), where cross-degrees exceed $W_5$3 times the opposite part size and internal degrees are bounded by the wheel parameter.

Closing the arguments

In the split regime for both theorems, the set $W_5$4 of neighbors of the Perron maximizer with coordinates exceeding $W_5$5 is bounded: for $W_5$6, inclusion–exclusion over common neighborhoods forces $W_5$7; for $W_5$8, a $W_5$9 would otherwise be embedded in $m_0$0, giving $m_0$1. The residual identity then shows $m_0$2 is forced (smaller values make the surplus negative since $m_0$3 eventually), and a further claim shows missing edges within $m_0$4 are incompatible with $m_0$5 because each non-edge contributes more than the deficit $m_0$6 it creates. Equality analysis collapses the structure to $m_0$7 with $m_0$8.

In the balanced regime, the two cases diverge. For $m_0$9, internal maximum degree at most one on each side suffices, and Proposition 3 applies directly. For $W_5$0, the argument is quantitative: writing $W_5$1, $W_5$2, and $W_5$3, a Rayleigh-quotient estimate gives $W_5$4 where $W_5$5. Since $W_5$6 from the internal-degree bound, a case analysis over $W_5$7 shows $W_5$8 unless $W_5$9. But then some vertex has internal degree exactly $G$0, its neighborhood in $G$1 has a common intersection $G$2 of size $G$3, and $G$4 forces two disjoint edges in $G$5 — which together with the $G$6 internal neighbors completes a $G$7 inside a single neighborhood, producing the forbidden wheel. Thus only the split configuration survives.

Limitations and open questions

The results are asymptotic: both theorems hold only for $G$8 beyond an unspecified threshold $G$9, and no effective bound on $W_{2k+1}=K_1\vee C_{2k}$00 is given. The equality characterization in Theorem 2 requires $W_{2k+1}=K_1\vee C_{2k}$01; when this divisibility fails, equality is unattainable, and the paper leaves open the determination of the exact maximum in each nonzero residue class, noting only that threshold graphs obtained from $W_{2k+1}=K_1\vee C_{2k}$02 by bounded defect are natural candidates. The dense-core constants are non-explicit, and the stability theorem invoked is itself qualitative in its edit-distance bound.

Conclusion

The paper settles the remaining Yu–Li–Peng conjecture by demonstrating that the $W_{2k+1}=K_1\vee C_{2k}$03 and general odd-wheel problems share a common dense-core/stability skeleton but differ in terminal rigidity: a matching can persist inside each side of a balanced complete bipartite graph without creating a $W_{2k+1}=K_1\vee C_{2k}$04 in any neighborhood, whereas the $W_{2k+1}=K_1\vee C_{2k}$05 obstruction eliminates all bounded-ratio configurations for $W_{2k+1}=K_1\vee C_{2k}$06. The combination of edge-spectral stability, dense-core reduction, and residual-function bookkeeping provides a template likely applicable to other fixed-size spectral problems for degenerate forbidden subgraphs.