Spread Number in Diverse Mathematical Domains

• Spread number is a parameter that quantifies dispersion by measuring vertex occurrences across bags in tree decompositions, linking local multiplicity with global treewidth.
• Trade-off theorems establish that for tree decompositions, the extreme spread-to-width constant lies between 2 and 3, using separator and recursive marking techniques.
• Beyond decompositions, spread number appears in branching models, spectral graph theory, and algebra, capturing asymptotic type proportions, eigenvalue gaps, and generator growth.

In contemporary mathematical and applied research, the expression spread number is not a universally standardized term. In the recent literature on tree decompositions, the underlying paper explicitly states that it “does not introduce a new symbol called ‘spread number,’” but instead studies extremal parameters derived from the spread of vertices across bags of a decomposition (Bodlaender et al., 7 Jan 2026). In other areas, closely related terminology denotes the asymptotic fraction of projected types in branching spread models, where “spread number” is also called “spread rate” (Ban et al., 2 Jan 2025), while spectral graph theory and matrix analysis use spread for the gap between extremal eigenvalues or, more generally, the diameter of the spectrum (Liu et al., 2024, Drnovšek, 2013). The term therefore refers not to a single invariant across mathematics, but to a family of quantities that measure extent, dispersion, or asymptotic prevalence in structures as diverse as tree decompositions, ideals, graphs, matrices, and epidemic processes.

1. Tree-decomposition spread as a recent core usage

Let $G$ be a graph and

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$

a tree-decomposition of $G$. In this setting, the spread of a vertex $v$ is

$$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$

The maximum spread of the decomposition is

$$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$

and the average spread is

$$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$

These definitions isolate how many bags a vertex occupies, or on average how many vertex-occurrences are present across the decomposition (Bodlaender et al., 7 Jan 2026).

The same paper formulates two extremal parameters. One is the infimum $c$ such that there is $c'$ with every graph $G$ of treewidth $\bigl(T,(B_x)_{x\in V(T)}\bigr)$0 admitting a decomposition of width $\bigl(T,(B_x)_{x\in V(T)}\bigr)$1 and $\bigl(T,(B_x)_{x\in V(T)}\bigr)$2. The other is the infimum $\bigl(T,(B_x)_{x\in V(T)}\bigr)$3 such that there is $\bigl(T,(B_x)_{x\in V(T)}\bigr)$4 with every graph admitting a decomposition of width $\bigl(T,(B_x)_{x\in V(T)}\bigr)$5 and $\bigl(T,(B_x)_{x\in V(T)}\bigr)$6 (Bodlaender et al., 7 Jan 2026).

This usage is structurally different from spectral or probabilistic notions of spread. Here the quantity is combinatorial and local-to-global: a vertex-level multiplicity constraint is compared against the global width parameter $\bigl(T,(B_x)_{x\in V(T)}\bigr)$7. A plausible implication is that “spread number” in this context is best understood as shorthand for one of these extremal spread-versus-width constants, rather than as a separately defined invariant.

2. Trade-off theorems for width and spread

The principal results pin down the first extremal parameter to the interval $\bigl(T,(B_x)_{x\in V(T)}\bigr)$8. Theorem 3.1 states: for every real $\bigl(T,(B_x)_{x\in V(T)}\bigr)$9 and every constant $G$0, there exists a graph $G$1 of treewidth $G$2 such that in every tree-decomposition of width at most $G$3 some vertex $G$4 has

$G$5

Thus $G$6 is necessary (Bodlaender et al., 7 Jan 2026).

Theorem 3.2 provides the complementary upper result: for each $G$7 there is an explicit $G$8 such that every graph $G$9 of treewidth $v$0 has a tree decomposition of width at most

$v$1

in which each vertex $v$2 has

$v$3

More specifically,

$v$4

where $v$5 (Bodlaender et al., 7 Jan 2026).

For average spread, Theorem 4.1 shows that the infimum is exactly $v$6: for every real $v$7 there exists $v$8 such that every graph $v$9 of treewidth $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$0 admits a tree-decomposition of width at most $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$1 and average spread

$$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$2

The discussion summarizes these results as follows: if one demands width $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$3 and a uniform bound $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$4, then $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$5 cannot be below $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$6 but any $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$7 suffices; if one measures storage by average spread, then the infimum of achievable average spread is exactly $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$8, while width remains $$\text{sp}_{\mathcal D}(v)\;=\;\bigl|\{\,x\in V(T):v\in B_x\}\bigr|\,.$$9 (Bodlaender et al., 7 Jan 2026).

The interval $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$0 therefore marks the unresolved zone for the true spread-versus-width constant, and the authors conjecture it is $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$1. This is the sharpest recent characterization in the supplied corpus of any quantity plausibly called a spread number.

3. Mechanisms behind the tree-decomposition bounds

The lower bound for $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$2 uses a “diagonal-grid $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$3 one-vertex” example $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$4, in which a special vertex $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$5 forces large spread once width $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$6 is imposed. A planar-separator-type argument, via a version of the Hex theorem, shows that any separator bag of size $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$7 must contain a long north–south path in each of many columns. By a careful “line of separators” along the path in the decomposition, one obtains

$$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$8

which contradicts any attempt to bound spread uniformly by a constant multiple of $$\max\text{‐spread}(\mathcal D) \;=\;\max_{v\in V(G)}\text{sp}_{\mathcal D}(v)\,,$$9 (Bodlaender et al., 7 Jan 2026).

The upper bound for $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$0 is based on a recursive separator construction together with a new $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$1-marking device. The key lemma states that if a rooted decomposition admits a $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$2-marking

$$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$3

satisfying a small monotonicity/happy-vertex property, then

$$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$4

The construction chooses, at each recursive step, only a $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$5-fraction of separator vertices to “pay” by adding a fresh neighbour, thereby decrementing their marking. This keeps each bag of size $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$6 while preserving a valid $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$7-marking, after which the lemma yields the desired spread bound (Bodlaender et al., 7 Jan 2026).

For average spread $$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$8, the proof begins with a nice width-$$\overline{\text{spread}(\mathcal D)} \;=\;\frac1{|V(G)|}\sum_{v\in V(G)}\text{sp}_{\mathcal D}(v) \;=\;\frac1{|V(G)|}\sum_{x\in V(T)}|B_x|\,.$$9 decomposition and partitions the decomposition tree into subtrees of size between $c$0 and $c$1. Each block is contracted to a single bag. The argument then shows that in each new bag at least a $c$2-fraction of its vertices are “new,” giving

$c$3

As $c$4 grows, this tends to $c$5 (Bodlaender et al., 7 Jan 2026).

These proof schemes clarify that the tree-decomposition notion of spread is not merely a counting device. It is tightly linked to separator geometry, recursive decomposition, and amortized accounting over bags.

4. Spread number as spread rate in projected branching models

In projected spread models, the terminology is explicit: the paper describes the spread number as also called the spread rate (Ban et al., 2 Jan 2025). The framework begins with hidden types $c$6, explicit types $c$7, and a projection $c$8. From an initial pattern $c$9, one constructs an infinite labeling $c'$0 on the regular $c'$1-ary tree and then projects to an $c'$2-labeling

$c'$3

Given positive integers $c'$4, with $c'$5, and the shell

$c'$6

the number of occurrences of $c'$7 is

$c'$8

The spread number is then defined by

$c'$9

provided the limit exists (Ban et al., 2 Jan 2025).

Under the primitivity hypotheses on the hidden spread model, one forms the nonnegative $G$0-matrix $G$1, lets $G$2 be its Perron–Frobenius eigenvalue, and normalizes the positive left eigenvector $G$3 by

$G$4

Then Theorem 2.2 gives

$G$5

and the limit is independent of the initial hidden type $G$6 (Ban et al., 2 Jan 2025).

In the random setting, the same asymptotic proportion arises from a $G$7-type Galton–Watson process with mean offspring matrix $G$8. Under mild moment conditions,

$G$9

and therefore the lumped fractions converge almost surely to the same eigenvector sums. This makes the projected spread number a spectral composition parameter rather than an outbreak threshold. The paper separately notes that $\bigl(T,(B_x)_{x\in V(T)}\bigr)$00 is the usual epidemic threshold, while the components of $\bigl(T,(B_x)_{x\in V(T)}\bigr)$01 describe the long-run type-composition of the spread (Ban et al., 2 Jan 2025).

5. Spectral meanings of spread in matrices and graphs

In matrix analysis, the spread of a matrix is an eigenvalue-diameter. For $\bigl(T,(B_x)_{x\in V(T)}\bigr)$02 with eigenvalues $\bigl(T,(B_x)_{x\in V(T)}\bigr)$03, the spread is

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$04

Thus $\bigl(T,(B_x)_{x\in V(T)}\bigr)$05 is the diameter of $\bigl(T,(B_x)_{x\in V(T)}\bigr)$06 in the complex plane. Immediate properties include $\bigl(T,(B_x)_{x\in V(T)}\bigr)$07, $\bigl(T,(B_x)_{x\in V(T)}\bigr)$08 for nilpotent $\bigl(T,(B_x)_{x\in V(T)}\bigr)$09, and $\bigl(T,(B_x)_{x\in V(T)}\bigr)$10 (Drnovšek, 2013).

For nonnegative matrices in

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$11

one has $\bigl(T,(B_x)_{x\in V(T)}\bigr)$12, and if exactly $\bigl(T,(B_x)_{x\in V(T)}\bigr)$13 diagonal entries vanish then

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$14

In the special class $\bigl(T,(B_x)_{x\in V(T)}\bigr)$15 consisting of matrices with exactly two distinct eigenvalues, Drnovšek proves

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$16

and this lower bound is best possible (Drnovšek, 2013).

In graph-related symmetric matrices, spread becomes the difference between the largest and smallest eigenvalue. For the $\bigl(T,(B_x)_{x\in V(T)}\bigr)$17-matrix

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$18

with ordered eigenvalues

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$19

the $\bigl(T,(B_x)_{x\in V(T)}\bigr)$20-spread is

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$21

Nikiforov, Liu, Fan, and Wang determine that for all sufficiently large $\bigl(T,(B_x)_{x\in V(T)}\bigr)$22, among all connected $\bigl(T,(B_x)_{x\in V(T)}\bigr)$23-vertex graphs, the quantity

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$24

is uniquely maximized by the kite graph $\bigl(T,(B_x)_{x\in V(T)}\bigr)$25, under $\bigl(T,(B_x)_{x\in V(T)}\bigr)$26, $\bigl(T,(B_x)_{x\in V(T)}\bigr)$27, and $\bigl(T,(B_x)_{x\in V(T)}\bigr)$28 (Liu et al., 2024).

A parallel construction appears for generalized distance matrices: $\bigl(T,(B_x)_{x\in V(T)}\bigr)$29 with spread

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$30

For transmission-regular graphs,

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$31

and several lower and upper bounds are given in terms of the Wiener index, Frobenius norm, transmission spread, clique number, and independence number (Ganie et al., 2019).

These spectral usages share with the tree-decomposition usage the general idea of “extent,” but the measured object is entirely different: the support of a vertex across bags in one case, and the extremal eigenvalue gap in the other.

6. Related notions of spread in algebra and epidemic theory

In commutative algebra, the symbolic analytic spread of an ideal $\bigl(T,(B_x)_{x\in V(T)}\bigr)$32 in a Noetherian local ring $\bigl(T,(B_x)_{x\in V(T)}\bigr)$33 is defined via the symbolic powers

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$34

and the minimal number of generators $\bigl(T,(B_x)_{x\in V(T)}\bigr)$35 by

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$36

This quantity compares with the ordinary analytic spread $\bigl(T,(B_x)_{x\in V(T)}\bigr)$37, and one always has

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$38

Under suitable Noetherianity and depth hypotheses, the paper derives bounds such as

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$39

and

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$40

(Dao et al., 2019). Here “spread” measures asymptotic generator growth, not spatial or spectral dispersion.

In epidemic models, the dominant scalar is often not called spread number but rather the basic reproduction number. For multi-type random contact graphs, the next-generation matrix $\bigl(T,(B_x)_{x\in V(T)}\bigr)$41 has entries

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$42

or in the paper’s normalization

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$43

and the threshold parameter is

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$44

The condition $\bigl(T,(B_x)_{x\in V(T)}\bigr)$45 is equivalent to supercritical branching and the emergence of a giant connected component, whereas $\bigl(T,(B_x)_{x\in V(T)}\bigr)$46 yields only small outbreaks (Minzer et al., 2021).

For SIS dynamics on interconnected directed networks, linearization at the disease-free equilibrium leads to a next-generation matrix $\bigl(T,(B_x)_{x\in V(T)}\bigr)$47, and again

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$48

The disease-free equilibrium is globally asymptotically stable if and only if $\bigl(T,(B_x)_{x\in V(T)}\bigr)$49, and for $\bigl(T,(B_x)_{x\in V(T)}\bigr)$50 there is a unique endemic equilibrium that is globally asymptotically stable in the interior of the feasible region (Jia et al., 2017).

A common misconception is that any “spread number” in epidemic work should mean $\bigl(T,(B_x)_{x\in V(T)}\bigr)$51. The supplied literature distinguishes them. In (Ban et al., 2 Jan 2025), the spread number is an asymptotic type proportion, while in (Minzer et al., 2021) and (Jia et al., 2017), the central scalar is the spectral-radius threshold $\bigl(T,(B_x)_{x\in V(T)}\bigr)$52.

7. Conceptual synthesis and current status

Across the supplied literature, “spread number” has no single canonical meaning. In tree decompositions, the phrase naturally points to extremal constants governing how spread

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$53

can be controlled relative to width and degree; the current best bounds place the critical width multiplier in $\bigl(T,(B_x)_{x\in V(T)}\bigr)$54, conjecturally $\bigl(T,(B_x)_{x\in V(T)}\bigr)$55, and establish average-spread infimum $\bigl(T,(B_x)_{x\in V(T)}\bigr)$56 (Bodlaender et al., 7 Jan 2026). In projected spread models, the term is formalized as a limit frequency

$\bigl(T,(B_x)_{x\in V(T)}\bigr)$57

computed by Perron–Frobenius eigenvector weights (Ban et al., 2 Jan 2025). In matrix and graph spectral theory, spread is an eigenvalue gap or spectral diameter (Drnovšek, 2013, Liu et al., 2024, Ganie et al., 2019). In commutative algebra, symbolic analytic spread tracks asymptotic growth of generators of symbolic powers (Dao et al., 2019).

What unifies these meanings is a shared measurement principle: each spread quantity records how widely some mathematical object is distributed across an ambient structure. The object may be a graph vertex across bags, an observable type across a branching tree, the spectrum across the complex plane, or the generators of symbolic powers across degree. This suggests that “spread number” functions less as a fixed term of art than as a domain-dependent label for an extremal or asymptotic extent parameter.