Spectral Booksize Bounds in Graph Theory

• The paper establishes quantitative relationships that link the largest eigenvalues of graph matrices with the maximal collection of triangles sharing a common edge, achieving tight thresholds like (1/3)√m.
• Advanced techniques such as Rayleigh quotient and Perron vector analysis are employed alongside combinatorial methods to derive supersaturation bounds and classify extremal graph constructions.
• These spectral bounds not only characterize forbidden subgraphs but also extend to generalized K₍r₊₁₎-book and cycle problems, offering deep insights and directing future research in extremal graph theory.

A spectral booksize bound provides a quantitative relationship between the largest eigenvalues of graph matrices (typically adjacency or signless Laplacian) and the maximal collection of triangles sharing a common edge—classically known as the booksize of a graph. Recent research unifies extremal graph theory, spectral inequalities, and supersaturation principles, yielding sharp, often tight, constraints relating eigenvalue extremality with booksize and generalizations to $K_{r+1}$-books, cycles, and more complex local structures.

1. Definitions and Classical Context

Let $G = (V, E)$ be a simple graph with $|V| = n$ vertices and $|E| = m$ edges. The adjacency matrix $A(G)$ and degree matrix $D(G)$ underlie two principal spectral invariants:

• The spectral radius $\rho(G)$ is the largest eigenvalue of $A(G)$.
• The Q-spectral radius $q(G)$ is the largest eigenvalue of the signless Laplacian $Q(G) = D(G) + A(G)$.

A book of size $G = (V, E)$0 ($G = (V, E)$1) is the union of $G = (V, E)$2 triangles sharing a common edge (the "spine"). The booksize $G = (V, E)$3 is the largest $G = (V, E)$4 for which $G = (V, E)$5 contains $G = (V, E)$6 as a subgraph; equivalently, it is the maximum number of triangles sharing a common edge in $G = (V, E)$7.

Spectral booksize bounds specify, for various spectral thresholds or forbidden subgraphs, the maximal or minimal possible booksize in $G = (V, E)$8, often with extremal graphs identified as achieving equality.

2. Supersaturation and Nosal-Type Bounds

The foundational result of Nosal asserts that if $G = (V, E)$9, then $|V| = n$0 contains a triangle. Extensions to booksize are provided in (Li et al., 20 Aug 2025, Zhai et al., 15 Jan 2026):

• Triangle-Book Supersaturation: If $|V| = n$1 is an $|V| = n$2-edge graph with $|V| = n$3, then $|V| = n$4 (Li et al., 20 Aug 2025). This is tight up to constant factors, as extremal constructions (clique–star join, blown-up prism) show $|V| = n$5 cannot exceed $|V| = n$6.
• Improved Bound for Nosal Graphs: Every Nosal graph ($|V| = n$7) satisfies the sharper $|V| = n$8 (Zhai et al., 15 Jan 2026). This constant is optimal up to a factor of $|V| = n$9, as demonstrated by the blown-up triangular prism $|E| = m$0.
• Generalized $|E| = m$1-book Bounds: For $|E| = m$2-edge $|E| = m$3 with $|E| = m$4, there exists an edge in at least $|E| = m$5 copies of $|E| = m$6 (i.e., $|E| = m$7) (Li et al., 20 Aug 2025).

These theorems confirm the conjectures of Nikiforov and Li–Liu–Zhang for the minimal booksize in graphs at the Nosal threshold and establish that such supersaturation is a sharp phenomenon in spectral extremal theory.

3. Spectral Extremal Structure and Forbidden Subgraphs

Spectral booksize bounds also characterize extremal graphs for forbidden subgraphs such as books and bipartite graphs, relating spectral radius thresholds to the existence or exclusion of $|E| = m$8:

• Spectral Characterization for $|E| = m$9-free Graphs: For any $A(G)$0 and $A(G)$1 $A(G)$2-free with $A(G)$3, $A(G)$4, with equality if and only if $A(G)$5 is complete bipartite (Zhai et al., 15 Jan 2026).
• Signless Laplacian Upper Bounds: If $A(G)$6 is $A(G)$7-free, order $A(G)$8, maximum degree $A(G)$9, then

$D(G)$0

(equality if and only if $D(G)$1 is strongly regular with parameters $D(G)$2) (Kong et al., 2016).

For $D(G)$3-free graphs ($D(G)$4), this specializes to the sharpest possible spectral bound given local control on triangle multiplicities.

• Non-Bipartite Book-Free Extremals: For large $D(G)$5 and $D(G)$6 non-bipartite, $D(G)$7-free,

$D(G)$8

unless $D(G)$9 is $\rho(G)$0 (obtained by adding one edge to one side of $\rho(G)$1), with $\rho(G)$2 (Zhai et al., 15 Jan 2026).

In each case, extremality coincides with classical structures: complete bipartite graphs for the $\rho(G)$3-edge threshold, strongly regular graphs for Laplacian bounds, and $\rho(G)$4 for non-bipartite forbidden subgraphs.

4. Turán-Type Spectral Extremal Results and Cycle Consequences

The spectral analogues of the Erdős–Simonovits–Turán theorems for books and cycles are established as follows:

• Spectral Critical Edge Theorem for Books: For $\rho(G)$5, any graph with $\rho(G)$6 contains $\rho(G)$7, with $\rho(G)$8 the complete bipartite graph with parts as equal as possible—matching the classical extremal structure (Zhai et al., 2021).
• Constant in Booksize Lower Bound: Explicitly, if $\rho(G)$9 and $A(G)$0, then $A(G)$1, improving toward but not reaching the Erdős bound ($A(G)$2).
• Extensions to Cycles and $A(G)$3-Graphs: The same spectral extremal method shows that for $A(G)$4 with $A(G)$5, $A(G)$6 contains cycles of every length $A(G)$7 (Zhai et al., 2021).

The constants (e.g., $A(G)$8) in these spectral booksize thresholds arise from delicate combinatorial and spectral inequalities involving neighborhood partitioning and Perron vector analysis near spectral extremality.

5. Proof Techniques: Eigenvector Analysis and Weighted Blowups

The proofs of spectral booksize bounds universally rely on advanced spectral techniques:

• Rayleigh Quotient & Perron Vector Analysis: By maximizing the quadratic form $A(G)$9 or $q(G)$0 at the Perron eigenvector and optimizing over neighborhood decompositions, global extremal properties are reduced to local combinatorial constraints.
• Weighted Edwards–Khadžiivanov–Nikiforov Lemmas: Generalizations to weighted settings, particularly for the triangle-book and higher joint distributions, permit passing from eigenvalue thresholds to explicit substructure supersaturation.
• Structural Dichotomies: Lemmas asserting that spectral extremal graphs must approximate bipartite or strongly regular forms, often via maximum degree constraints or local independence.
• Zarankiewicz-Type Counting and Convexity: Extremal forbidden subgraph bounds are derived via bipartite reductions and convexity properties of spectral parameters.
• Cauchy–Schwarz and Concentration of Measure: Essential for passing from average weight/eigenvector-based conclusions to existence of large book subgraphs.

These methods yield both existential and quantitative guarantees for the presence of large books and generalized joints, tightly matching or bounding possible extremal constructions.

6. Extremal Constructions and Sharpness

Explicit constructions show that the spectral booksize bounds are optimal up to constant factors:

Construction	$q(G)$1 (edges)	$q(G)$2 and $q(G)$3
$q(G)$4	$q(G)$5, $q(G)$6	$q(G)$7, $q(G)$8
$q(G)$9 + chord	$Q(G) = D(G) + A(G)$0, $Q(G) = D(G) + A(G)$1	$Q(G) = D(G) + A(G)$2, $Q(G) = D(G) + A(G)$3
Blown-up prism $Q(G) = D(G) + A(G)$4	$Q(G) = D(G) + A(G)$5	$Q(G) = D(G) + A(G)$6, $Q(G) = D(G) + A(G)$7

All these confirm that the order $Q(G) = D(G) + A(G)$8 in supersaturation bounds is tight, and constants cannot be improved beyond $Q(G) = D(G) + A(G)$9 under current methods (Li et al., 20 Aug 2025, Zhai et al., 15 Jan 2026).

7. Open Problems and Future Directions

Several important open questions remain:

• Determination of the exact optimal constants in spectral booksize thresholds, with the best known lying between $G = (V, E)$00 and $G = (V, E)$01 for Nosal graphs. Is $G = (V, E)$02 optimal?
• Extending spectral booksize theory to smaller $G = (V, E)$03 or alternative forbidden configurations, and to signless Laplacian or normalized Laplacian eigenvalues.
• Further connection between spectral supersaturation and classical extremal combinatorics, especially in determining the structural stability of near-extremal graphs.
• Enhancement of cycle-spanning constants and the unification of spectral extremal principles for books and more complex local substructures.

Current research continues to bridge extremal graph theory, spectral analysis, and combinatorial enumeration, expanding both the toolkit and the scope of spectral booksize bounds (Kong et al., 2016, Zhai et al., 2021, Li et al., 20 Aug 2025, Zhai et al., 15 Jan 2026).