Linear Jaco Graphs

• Linear Jaco graphs are indexed graphs defined by f(x)=mx+c, where each vertex’s degree and arc set are determined by explicit algebraic rules.
• They form acyclic, finite or infinite digraphs with arcs from lower to higher indexed vertices, showcasing contiguous in-neighborhood intervals.
• Their construction allows closed-form solutions in special cases and underpins studies in degree formulas, arc counts, and invariant properties.

Linear Jaco graphs are a family of indexed graphs generated from a canonical infinite root digraph and then truncated to finite order. In the general linear form, the defining parameter is a linear function $f(x)=mx+c$ with $m\in\mathbb{N}$ and $c\in\mathbb{N}_0$; vertices are $v_1,v_2,\dots$, arcs always point from lower to higher indices, and the infinite graph is constrained so that the directed degree of $v_k$ is $d(v_k)=mk+c$. The special case $f(x)=x$ yields the order-$1$ Jaco graphs $J_n(1)$, which became the principal test case for later work on degree formulas, arc counts, Jaconian vertices, Hope subgraphs, Fibonacci–Zeckendorf structure, and several invariant-theoretic extensions (Kok et al., 2015, Kok et al., 2014, Kok et al., 2014).

1. Definition and formal variants

For $f(x)=mx+c$, the infinite linear Jaco digraph $m\in\mathbb{N}$0 is defined by

$m\in\mathbb{N}$1

with

$m\in\mathbb{N}$2

and

$m\in\mathbb{N}$3

Equivalently, the largest reachable head from $m\in\mathbb{N}$4 is $m\in\mathbb{N}$5, so the out-neighborhood of $m\in\mathbb{N}$6 is the consecutive block $m\in\mathbb{N}$7. The finite graph $m\in\mathbb{N}$8 is obtained by “lobbing off” all vertices $m\in\mathbb{N}$9 with $c\in\mathbb{N}_0$0 and all incident arcs, hence

$c\in\mathbb{N}_0$1

and

$c\in\mathbb{N}_0$2

The four fundamental properties recorded for $c\in\mathbb{N}_0$3 are: the vertex set is $c\in\mathbb{N}_0$4; every head $c\in\mathbb{N}_0$5 has only tails $c\in\mathbb{N}_0$6 with $c\in\mathbb{N}_0$7; if $c\in\mathbb{N}_0$8 is the smallest tail to $c\in\mathbb{N}_0$9, then all $v_1,v_2,\dots$0 with $v_1,v_2,\dots$1 are also tails to $v_1,v_2,\dots$2; and $v_1,v_2,\dots$3 (Kok et al., 2015).

This framework extends earlier order-$v_1,v_2,\dots$4 Jaco graphs $v_1,v_2,\dots$5, where the defining inequality is

$v_1,v_2,\dots$6

and the infinite degree law is $v_1,v_2,\dots$7. The order-$v_1,v_2,\dots$8 case $v_1,v_2,\dots$9 therefore satisfies

$v_k$0

with $v_k$1 (Kok et al., 2014, Kok et al., 2014).

A later experimental study uses the notation $v_k$2 for the underlying simple, connected, undirected linear Jaco graphs, and explicitly notes that the parameter $v_k$3 is only a nominal label there; the directed construction from a linear function $v_k$4 is not used in that paper (Kok, 22 Jul 2025).

2. Degree laws, tail intervals, and the auxiliary sequence

A central device in the linear theory is the auxiliary sequence $v_k$5, defined by

$v_k$6

It is well-defined, ascending, and satisfies

$v_k$7

In $v_k$8 one then has

$v_k$9

together with

$d(v_k)=mk+c$0

and the slow-growth rule

$d(v_k)=mk+c$1

For a fixed head $d(v_k)=mk+c$2, the minimal tail index is exactly

$d(v_k)=mk+c$3

and

$d(v_k)=mk+c$4

Thus every in-neighborhood is a contiguous index interval (Kok et al., 2015).

Quantity	Formula
Total degree in $d(v_k)=mk+c$5	$d(v_k)=mk+c$6
Auxiliary sequence	$d(v_k)=mk+c$7
In-degree	$d(v_k)=mk+c$8
Out-degree	$d(v_k)=mk+c$9
Minimal tail to $f(x)=x$0	$f(x)=x$1

In the finite truncation $f(x)=x$2, the in-degree of $f(x)=x$3 is inherited from the infinite construction whenever only lower indices are involved, while out-degree is truncated at the boundary. The explicit finite formula is

$f(x)=x$4

so

$f(x)=x$5

Equality holds whenever the infinite out-neighborhood of $f(x)=x$6 fits entirely inside $f(x)=x$7, that is, whenever

$f(x)=x$8

The order-$f(x)=x$9 theory has the analogous sequence $1$0 and degree identity

$1$1

with $1$2 (Kok et al., 2015, Kok et al., 2014).

3. Finite structure, adjacency matrix, and extremal vertices

Because every arc goes from lower to higher index, both $1$3 and each $1$4 are acyclic directed graphs with natural topological ordering $1$5. In matrix form, when rows and columns are indexed by vertex order, the adjacency matrix is upper triangular, and each column $1$6 has a contiguous block of $1$7’s from row $1$8 to row $1$9. This columnwise interval structure is one of the characteristic signatures of linear Jaco graphs (Kok et al., 2015, Kok et al., 2014).

The underlying undirected graph $J_n(1)$0 is connected for $J_n(1)$1. The paper also isolates disconnected limiting families: if $J_n(1)$2 and $J_n(1)$3, then

$J_n(1)$4

while $J_n(1)$5 is the null graph. For finite linear Jaco graphs, the minimum degree satisfies

$J_n(1)$6

and consecutive degree differences obey

$J_n(1)$7

In the order-$J_n(1)$8 case, the underlying undirected graph also contains the monotone Hamiltonian path

$J_n(1)$9

since $f(x)=mx+c$0 is always present (Kok et al., 2015, Kok et al., 2014).

Vertices attaining $f(x)=mx+c$1 are the Jaconian vertices, denoted $f(x)=mx+c$2; the least-indexed such vertex is the prime Jaconian vertex. If $f(x)=mx+c$3 is prime Jaconian, then the Hope subgraph $f(x)=mx+c$4 is the complete subgraph induced by $f(x)=mx+c$5. A useful rigidity statement is that if the prime Jaconian vertex $f(x)=mx+c$6 achieves $f(x)=mx+c$7 in $f(x)=mx+c$8, then all vertices $f(x)=mx+c$9 with $m\in\mathbb{N}$00 also satisfy $m\in\mathbb{N}$01, and $m\in\mathbb{N}$02 form a clique in the underlying graph. In the order-$m\in\mathbb{N}$03 theory, the Jaconian set has cardinality at most $m\in\mathbb{N}$04 (Kok et al., 2015, Kok et al., 2014).

The literature also gives criteria for identifying the prime Jaconian vertex. For example, if $m\in\mathbb{N}$05 is the smallest index with $m\in\mathbb{N}$06 and $m\in\mathbb{N}$07 exists, then $m\in\mathbb{N}$08 is prime Jaconian. In the order-$m\in\mathbb{N}$09 setting, $m\in\mathbb{N}$10 is prime Jaconian in $m\in\mathbb{N}$11 if and only if $m\in\mathbb{N}$12 for all $m\in\mathbb{N}$13 (Kok et al., 2015, Kok et al., 2014).

4. Enumeration, edge counts, and arithmetic specializations

Arc counting is explicit only in special regimes. For $m\in\mathbb{N}$14 and $m\in\mathbb{N}$15,

$m\in\mathbb{N}$16

because $m\in\mathbb{N}$17 is complete. More generally, if $m\in\mathbb{N}$18, then

$m\in\mathbb{N}$19

since the Hope subgraph contributes $m\in\mathbb{N}$20 arcs and the remaining contribution comes from the first $m\in\mathbb{N}$21 vertices. For $m\in\mathbb{N}$22, the recursive update

$m\in\mathbb{N}$23

holds when $m\in\mathbb{N}$24 is the prime Jaconian vertex of $m\in\mathbb{N}$25. A further special closed form is given at

$m\in\mathbb{N}$26

where

$m\in\mathbb{N}$27

The same paper states that finding a closed formula for $m\in\mathbb{N}$28 for general $m\in\mathbb{N}$29 is the chief open problem of the linear theory (Kok et al., 2015).

In the order-$m\in\mathbb{N}$30 case, the edge-count note gives three exact formulas for $m\in\mathbb{N}$31. The basic recursion is

$m\in\mathbb{N}$32

Equivalently,

$m\in\mathbb{N}$33

and for $m\in\mathbb{N}$34,

$m\in\mathbb{N}$35

A third formula decomposes $m\in\mathbb{N}$36 at a Jaconian index $m\in\mathbb{N}$37 when $m\in\mathbb{N}$38 (Kok et al., 2014).

These counting formulas are tied to Fibonacci arithmetic through Bettina’s Theorem. If

$m\in\mathbb{N}$39

is the Zeckendorf representation of $m\in\mathbb{N}$40, then in $m\in\mathbb{N}$41

$m\in\mathbb{N}$42

The same phenomenon appears in the general order-$m\in\mathbb{N}$43 theory as a Lucassian–Zeckendorf representation involving the generalized Lucas sequence $m\in\mathbb{N}$44; there

$m\in\mathbb{N}$45

For $m\in\mathbb{N}$46, the order-$m\in\mathbb{N}$47 formulas give $m\in\mathbb{N}$48 (Kok et al., 2014, Kok et al., 2014, Kok et al., 2014).

5. Order-$m\in\mathbb{N}$49 invariants, transforms, and associated graph families

The special case $m\in\mathbb{N}$50 supports a large invariant theory on the underlying undirected graph. The independence number is given recursively by the set

$m\in\mathbb{N}$51

and $m\in\mathbb{N}$52. The same paper proves

$m\in\mathbb{N}$53

where $m\in\mathbb{N}$54 is the prime Jaconian vertex. It also gives $m\in\mathbb{N}$55, states that $m\in\mathbb{N}$56 for $m\in\mathbb{N}$57, and proves

$m\in\mathbb{N}$58

for the murtage number (Kok et al., 2014).

Several later constructions retain the canonical Jaco orientation. For the Mycielski Jaco graph, the brush-number result is

$m\in\mathbb{N}$59

and the terminal brush positions after an optimal cleaning sequence form a brush centre of $m\in\mathbb{N}$60. For the competition graph, the characterization for $m\in\mathbb{N}$61 is

$m\in\mathbb{N}$62

where $m\in\mathbb{N}$63. The same paper introduces the grog number and gives the recursion

$m\in\mathbb{N}$64

when $m\in\mathbb{N}$65 is a Jaconian vertex of $m\in\mathbb{N}$66 (Kok et al., 2015, Kok et al., 2015).

Triangle structure is encoded by the primitive hole number on the underlying graph $m\in\mathbb{N}$67. If $m\in\mathbb{N}$68 is the prime Jaconian vertex of $m\in\mathbb{N}$69, then

$m\in\mathbb{N}$70

and for $m\in\mathbb{N}$71,

$m\in\mathbb{N}$72

The irregularity paper studies the underlying graphs $m\in\mathbb{N}$73 via

$m\in\mathbb{N}$74

and

$m\in\mathbb{N}$75

with recursive updates based on the prime Jaconian vertex (Kok et al., 2015, Kok, 2014).

6. Energy interpretation, experimental undirected model, and open directions

Finite Jaco-type graphs are also energy graphs in the sense that they are simple, directed, vertex-labeled graphs with $m\in\mathbb{N}$76 when $m\in\mathbb{N}$77 and at least one source vertex. In this framework, linear Jaco graphs arise from the sequence $m\in\mathbb{N}$78, so

$m\in\mathbb{N}$79

and $m\in\mathbb{N}$80 is the induced finite truncation. The graph has unique source $m\in\mathbb{N}$81 and primary sink $m\in\mathbb{N}$82. Black arcs are those in-arcs not lying on any minimum directed distance path from a source, and the Jaco-type Black Arc Algorithm determines all of them. For $m\in\mathbb{N}$83,

$m\in\mathbb{N}$84

with

$m\in\mathbb{N}$85

and total black energy

$m\in\mathbb{N}$86

joules. The same paper emphasizes that although $m\in\mathbb{N}$87 is a DAG, its underlying undirected graph may contain cycles, so black energy dissipation can occur (Kok et al., 2016).

A distinct recent experimental program studies the undirected graphs $m\in\mathbb{N}$88 and reports verified Beatty-type formulas

$m\in\mathbb{N}$89

together with conjectures such as

$m\in\mathbb{N}$90

Further conjectures place unique $m\in\mathbb{N}$91-values, orders with $m\in\mathbb{N}$92, and canonical $m\in\mathbb{N}$93-set indices into Wythoff and Beatty families, while the paper marks them explicitly as conjectural rather than proved (Kok, 22 Jul 2025).

Open directions recur throughout the literature. The linear-function paper identifies the search for a closed formula for $m\in\mathbb{N}$94 for general $m\in\mathbb{N}$95 as the chief unresolved counting problem and proposes extensions to polynomial Jaco graphs $m\in\mathbb{N}$96 with

$m\in\mathbb{N}$97

Other stated problems include adjacency-matrix characterizations, directed-connectivity questions, a general Black Arc Algorithm, solid-subgraph connectivity in energy graphs, and proof or disproof of the recent Beatty/Wythoff conjectures for the undirected model (Kok et al., 2015, Kok et al., 2016, Kok, 22 Jul 2025).