Spectral Radius Order Research

• Spectral radius order is the study of comparing combinatorial, algebraic, and dynamical systems using the largest eigenvalue of associated matrices, tensors, or maps.
• It provides a unified framework for ordering graphs under structural constraints such as degree sequences, sizes, and forbidden subgraph conditions.
• This approach bridges extremal spectral graph theory with applications in equiangular lines and nonlinear Perron–Frobenius methods.

to=arxiv_search ्राjson {"^^^^^^^^3query3 radius order\"^^^^^^^^3 OR ti:\3^^^^^^^^"spectral radius order\"","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"}цҳауеитුanalysis to=arxiv_search 天天彩票怎么json code omitted because not possible? to=arxiv_search 日本一级特黄大片 {"^^^^^^^^3query3 radius order", "max_results": ^^^^3all:\3query3^^^^, "sort_by": "submittedDate", "sort_order": "descending"} Spectral radius order is the study of how combinatorial, algebraic, or dynamical objects are compared through the largest eigenvalue or spectral growth parameter of an associated matrix, tensor, or map. In graph theory, it commonly refers to ordering graphs or digraphs by adjacency or PRESERVED_PLACEHOLDER^^^^3query3^^^^-spectral radius under structural constraints such as degree sequence, size, diameter, rank, or forbidden subgraphs. In a more specialized recent sense, the spectral radius order of a real number PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^ is the minimum number of vertices in a graph whose adjacency spectral radius equals PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^ (&&&^^^^3query3^^^^&&&). Across these usages, the subject connects extremal spectral graph theory, nonlinear Perron–Frobenius theory, tensor spectra, and applications such as equiangular lines (&&&^^^^3all:\3^^^^&&&).

^^^^3all:\3^^^^. Definitions and principal frameworks

The basic graph-theoretic object is the spectral radius $\rho(G)$, the largest eigenvalue of the adjacency matrix of a graph $G$. For digraphs and mixed degree-adjacency models, one studies the $A_\alpha$ matrix

$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$

whose largest eigenvalue is denoted $\rho_\alpha(G)$ for graphs and $\lambda_\alpha(G)$ for digraphs; this interpolates between adjacency spectra and, at $\alpha=\tfrac12$, half the signless Laplacian spectral radius (&&&^^^^3 OR ti:\3^^^^&&&). In higher-order settings, one replaces matrices by tensors, such as the PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^-clique tensor with spectral radius PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^, or by order-preserving homogeneous maps on cones, where several nonlinear notions of spectral radius arise (Wang et al., 6 Oct 2025, Akian et al., 2011).

A distinct but related definition was introduced for real numbers. For PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^, the spectral radius order PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^ is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4

with PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5 if no such graph exists (&&&^^^^3query3^^^^&&&). This converts a realization problem into a minimal-order problem and makes “order” literal: it measures the smallest graph size that realizes a prescribed spectral radius.

These frameworks lead to different but compatible questions. One asks for exact extremal objects that maximize or minimize spectral radius in a fixed class, for sharp upper or lower bounds in terms of degrees or other combinatorial invariants, or for algebraic criteria determining when a number can appear as a graph spectral radius of finite order.

^^^^3 OR ti:\3^^^^. Degree-sequence order and sharp comparison bounds

A foundational direction orders graphs through degree data. For a simple connected graph PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6 of order PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7 with degree sequence PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8, a sharp family of upper bounds was given by

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9

for every integer PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^ (&&&^^^^3all:\3^^^^&&&). This bound generalizes earlier results of Stanley, Hong et al., and Shu and Wu, and it is sharp. Equality holds if and only if PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^ is regular, or there exists PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^ such that PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^ and PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4 (&&&^^^^3all:\3^^^^&&&).

An important structural point is that the sequence PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5 is not necessarily monotonic, so spectral radius order cannot in general be read off from a single truncation of the degree sequence. The tightest estimate is

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6

and the usefulness of the method lies precisely in choosing PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7 strategically (&&&^^^^3all:\3^^^^&&&). This shows that degree-sequence ordering is not a one-parameter phenomenon but a hierarchy of comparisons.

For bipartite graphs, degree-sequence comparison becomes genuinely two-sided. Sharp upper bounds in terms of the degree sequences on both parts were derived for bipartite PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8, and these were used to solve an extremal edge-deletion problem: if PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9, then among all subgraphs obtained from $\rho(G)$^^^^3query3^^^^ by deleting $\rho(G)$^^^^3all:\3^^^^ edges, the maximum spectral radius is attained when all deleted edges are incident to a single vertex in the partite set of order $\rho(G)$^^^^3 OR ti:\3^^^^ (Liu et al., 2014). The result gives a concrete spectral ordering principle: localized degree damage dominates dispersed damage.

These results also delimit common misunderstandings. Degree information can control $\rho(G)$3 very closely, but neither monotonicity in truncation index nor naive “more irregular means larger” rules are universally valid. The bounds are strongest when they exploit the full structure of the degree sequence rather than a single extremal degree.

3. Extremal orderings for constrained graph classes

A large part of spectral radius order is extremal classification under fixed parameters. For connected graphs with prescribed order $\rho(G)$4 and size $\rho(G)$5, recent work on minimum spectral radius addresses a question of Hong from ^^^^3all:\3^^^^993: whether minimizers must be almost regular, meaning $\rho(G)$6. This question is answered positively for various sparse and dense regimes, for most cases when $\rho(G)$7, and in many sporadic cases; in the explicit cases analyzed, the minimizers are regular or have two degrees differing by one (&&&^^^^3all:\3query3^^^^&&&). The same work records small counterexamples to monotonicity of spectral radius in degree irregularity, but these do not furnish minimizers in Hong’s sense (&&&^^^^3all:\3query3^^^^&&&).

At the opposite extremal end, the maximum spectral radius for connected graphs of given order and rank is uniquely attained by the Turán graph $\rho(G)$8, the complete $\rho(G)$9-partite graph with parts as equal as possible (&&&^^^^3all:\3 OR ti:\3^^^^&&&). This identifies spectral radius order with a classical extremal partition principle. For fixed order and size, threshold graphs occupy a similarly central role: each connected extremal graph is necessarily a threshold graph, and recent work develops lower and upper bounds for the spectral radius of connected threshold graphs via lazy walks (&&&^^^^3all:\33^^^^&&&). In adjacent extremal classification, the maximum spectral radius problem on the class $G$^^^^3query3^^^^ of connected graphs with $G$^^^^3all:\3^^^^ vertices and $G$^^^^3 OR ti:\3^^^^ edges is solved when $G$3 or $G$4, with the maximizer always among two explicit threshold families $G$5 and $G$6 (&&&^^^^3all:\3^^^^4&&&). For $G$7-spectral radius, connected maximizers with fixed order and size are again threshold graphs; for $G$8 and $G$9, the unique extremal graph is typically the quasi-star threshold graph $A_\alpha$^^^^3query3^^^^, with specific exceptional cases at $A_\alpha$^^^^3all:\3^^^^ (&&&^^^^3all:\3^^^^5&&&).

Spectral orderings have also been resolved for several constrained families. For graphs of order $A_\alpha$^^^^3 OR ti:\3^^^^ without the $A_\alpha$3-power of a Hamilton cycle, the unique graph of maximum spectral radius for $A_\alpha$4 is $A_\alpha$5 (&&&^^^^3all:\3^^^^6&&&). For graphs of order $A_\alpha$6 with given fractional matching number $A_\alpha$7, the extremal graphs for maximum spectral radius are explicit joins, with a phase split between complete graphs, join-plus-isolates constructions, and $A_\alpha$8 according to the range of $A_\alpha$9 relative to $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$^^^^3query3^^^^ (&&&^^^^3all:\3^^^^7&&&). For graphs of order $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$^^^^3all:\3^^^^ and diameter $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$^^^^3 OR ti:\3^^^^, the minimizers are exactly the open quipus

$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$3

and all members of this family have the same spectral radius (&&&^^^^3all:\3^^^^8&&&).

Taken together, these results show that spectral radius order is rarely governed by a single invariant. Clique concentration, near-regularity, threshold structure, and forbidden-subgraph geometry can each become decisive, depending on the constraint set.

4. Digraph orderings and $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$4-spectral comparisons

For strongly connected digraphs, exact orderings have been obtained at both the lower and upper ends. Among all strongly connected digraphs of order $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$5, the unique digraphs with the first four smallest adjacency spectral radii are

$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$6

in that order (&&&^^^^3all:\3^^^^9&&&). This gives one of the cleanest complete spectral orderings in the subject.

The $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$7 setting yields threshold-type comparison principles based on outdegree. For a strongly connected digraph $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$8, with

$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$9

two explicit ordering theorems were proved (&&&^^^^3 OR ti:\3^^^^&&&). First, if $\rho_\alpha(G)$^^^^3query3^^^^ and $\rho_\alpha(G)$^^^^3all:\3^^^^ are strongly connected digraphs with $\rho_\alpha(G)$^^^^3 OR ti:\3^^^^ vertices and $\rho_\alpha(G)$3 arcs, $\rho_\alpha(G)$4, and

$\rho_\alpha(G)$5

then

$\rho_\alpha(G)$6

Second, if $\rho_\alpha(G)$7,

$\rho_\alpha(G)$8

then

$\rho_\alpha(G)$9

and for $\lambda_\alpha(G)$^^^^3query3^^^^ this strengthens to $\lambda_\alpha(G)$^^^^3all:\3^^^^ (&&&^^^^3 OR ti:\3^^^^&&&). These theorems show that maximum outdegree orders the $\lambda_\alpha(G)$^^^^3 OR ti:\3^^^^-spectral radius only above explicit thresholds; below those thresholds, a simple outdegree comparison is not enough.

On the graph side, a related $\lambda_\alpha(G)$3-ordering phenomenon appears under branch balancing. For the graph $\lambda_\alpha(G)$4, obtained by attaching paths $\lambda_\alpha(G)$5 and $\lambda_\alpha(G)$6 to distinguished vertices $\lambda_\alpha(G)$7 and $\lambda_\alpha(G)$8, the inequality

$\lambda_\alpha(G)$9

holds whenever $\alpha=\tfrac12$^^^^3query3^^^^ and $\alpha=\tfrac12$^^^^3all:\3^^^^ (&&&^^^^3 OR ti:\3 OR ti:\3^^^^&&&). Thus, shifting one pendent edge from the longer path to the shorter path increases the $\alpha=\tfrac12$^^^^3 OR ti:\3^^^^-spectral radius. This balancing principle yields exact extremal graphs for fixed order and cut vertices, and for trees with fixed order and matching number (&&&^^^^3 OR ti:\3 OR ti:\3^^^^&&&).

A recurring theme is that spectral order in directed or $\alpha=\tfrac12$3-weighted settings remains explicit, but only after the relevant parameter regime is identified. Outdegree, pendent-path balance, and threshold structure provide order parameters, yet always with sharp hypotheses.

5. Spectral radius order as a realization invariant

The most literal use of the term is the invariant $\alpha=\tfrac12$4, the minimum order of a graph whose adjacency spectral radius is exactly $\alpha=\tfrac12$5 (&&&^^^^3query3^^^^&&&). If $\alpha=\tfrac12$6, then $\alpha=\tfrac12$7 must be a totally real algebraic integer and must satisfy

$\alpha=\tfrac12$8

for every algebraic conjugate $\alpha=\tfrac12$9 of PRESERVED_PLACEHOLDER_^^^^3all:\3query3query3^^^^ (&&&^^^^3query3^^^^&&&). These are necessary conditions.

For general PRESERVED_PLACEHOLDER_^^^^3all:\3query3all:\3^^^^, those conditions are not sufficient, but two large classes are completely characterized. If PRESERVED_PLACEHOLDER_^^^^3all:\3query3 OR ti:\3^^^^ is a quadratic algebraic integer, or if PRESERVED_PLACEHOLDER_^^^^3all:\3query33^^^^, then PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^4 if and only if PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^5 satisfies the above algebraic conditions (&&&^^^^3query3^^^^&&&). In the range PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^6, Kronecker’s theorem reduces the realizable values to the classical PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^7-type algebraic integers, realized by path graphs, with PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^8 accounting for PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^9 (&&&^^^^3query3^^^^&&&).

The paper also derives exact values for infinite families. If

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3query3^^^^

then

PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3all:\3^^^^

realized by the complete bipartite graph PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3 OR ti:\3^^^^ (&&&^^^^3query3^^^^&&&). This establishes a precise minimal-order formula for a nontrivial quadratic family.

The significance of PRESERVED_PLACEHOLDER_^^^^3all:\3all:\33^^^^ extends beyond graph realization. The concept plays a crucial role in the breakthrough work on equiangular lines due to Jiang, Tidor, Yao, Zhang, and Zhao, and the cited work states that for PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^4, finiteness of PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^5 controls the asymptotic size of the largest equiangular line systems with angle PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^6 (&&&^^^^3query3^^^^&&&). In this sense, spectral radius order becomes a bridge between algebraic graph spectra and extremal geometry.

A common misconception is that the necessary algebraic conditions should characterize all finite-order spectral radii. The recent results show this is true for quadratic algebraic integers and for PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^7, but not in general (&&&^^^^3query3^^^^&&&).

6. Higher-order, tensor, and nonlinear ordered extensions

The idea of ordering by spectral radius extends beyond matrices. For PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^8-order PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^9-tensors with PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3query3^^^^ ones, the maximum spectral radius satisfies

PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3all:\3^^^^

with equality if and only if PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3 OR ti:\3^^^^ for some integer PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\33^^^^ and all ones form a principal sub-tensor PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^4 (&&&3^^^^3all:\3^^^^&&&). This is the tensor analogue of a sharp extremal spectral ordering, and the exponent PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^5 makes the dependence on tensor order explicit.

For graphs, the high-order PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^6-clique spectral radius PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^7 generalizes adjacency spectral radius because PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^8 is the ordinary adjacency spectral radius (Wang et al., 6 Oct 2025). High-order spectral Erdős–Gallai theorems determine the extremal graphs maximizing PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^9 under forbidden long cycles or paths. For small PRESERVED_PLACEHOLDER_^^^^3all:\33query3^^^^, the maximizers are join-type graphs PRESERVED_PLACEHOLDER_^^^^3all:\33all:\3^^^^ or PRESERVED_PLACEHOLDER_^^^^3all:\33 OR ti:\3^^^^; for PRESERVED_PLACEHOLDER_^^^^3all:\333^^^^, the extremal graph becomes PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^4 (Wang et al., 6 Oct 2025). This introduces a threshold phenomenon in the order parameter PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^5: the spectral ordering changes regime when clique size dominates path or cycle structure.

In nonlinear ordered spaces, spectral order is tied to order-preserving homogeneous maps on cones. A Collatz–Wielandt type formula characterizes the spectral radius by

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^6

under normality and suitable continuity hypotheses, and under quasi-compactness this equals the maximal eigenvalue associated with an eigenvector in the cone (Akian et al., 2011). For bounded, equicontinuous families of order-preserving homogeneous maps on a polyhedral cone, the Berger–Wang formula

PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^7

holds, equating generalized and joint spectral radii; further boundedness criteria for the semigroup PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^8 are available under irreducibility or primitivity when PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^9 (Lins et al., 2 Sep 2025). These results do not order finite graphs, but they extend the same governing idea: growth rates of ordered systems can be compared through sharp spectral invariants.

A plausible implication is that spectral radius order is best viewed not as a single theorem but as a unifying program. In graphs it yields exact extremal structures, in algebraic realization it yields the invariant PRESERVED_PLACEHOLDER_^^^^3all:\3max_results3query3^^^^, and in tensors and cone maps it organizes higher-order or nonlinear growth by the same Perron–Frobenius logic (&&&^^^^3query3^^^^&&&, Lins et al., 2 Sep 2025).