---
title: Linear Jaco Graphs
url: https://www.emergentmind.com/topics/linear-jaco-graphs
type: topic
---

# Linear Jaco Graphs

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 [1506.06538] [1404.1714] [1404.0484].

## 1. Definition and formal variants

For $f(x)=mx+c$, the infinite linear Jaco digraph $J_\infty(f)$ is defined by
$$
V(J_\infty(f))=\{v_i:i\in\mathbb{N}\},
$$
with
$$
A(J_\infty(f))\subseteq \{(v_i,v_j):i<j\},
$$
and
$$
(v_i,v_j)\in A(J_\infty(f)) \iff (f(i)+i)-d^-(v_i)\ge j.
$$
Equivalently, the largest reachable head from $v_i$ is $j_{\max}(i)=(f(i)+i)-d^-(v_i)$, so the out-neighborhood of $v_i$ is the consecutive block $v_{i+1},\dots,v_{j_{\max}(i)}$. The finite graph $J_n(f)$ is obtained by “lobbing off” all vertices $v_t$ with $t>n$ and all incident arcs, hence
$$
V(J_n(f))=\{v_1,\dots,v_n\},
$$
and
$$
(v_i,v_j)\in A(J_n(f)) \iff i<j\le n \text{ and } (f(i)+i)-d^-(v_i)\ge j.
$$
The four fundamental properties recorded for $J_\infty(f)$ are: the vertex set is $\{v_i:i\in\mathbb{N}\}$; every head $v_j$ has only tails $v_i$ with $i<j$; if $v_k$ is the smallest tail to $v_j$, then all $v_\ell$ with $k<\ell<j$ are also tails to $v_j$; and $d(v_k)=mk+c=f(k)$ [1506.06538].

This framework extends earlier order-$a$ Jaco graphs $J_\infty(a)$, where the defining inequality is
$$
(v_i,v_j)\in E(J_\infty(a)) \iff (a+1)i-d^-(v_i)\ge j,
$$
and the infinite degree law is $d(v_k)=ak$. The order-$1$ case $J_\infty(1)$ therefore satisfies
$$
(v_i,v_j)\in E(J_\infty(1)) \iff 2i-d^-(v_i)\ge j,
$$
with $d(v_k)=k$ [1404.1714] [1404.0484].

A later experimental study uses the notation $J_n(x)$ for the underlying simple, connected, undirected linear Jaco graphs, and explicitly notes that the parameter $x$ is only a nominal label there; the directed construction from a linear function $f(x)=mx+c$ is not used in that paper [2507.16500].

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

A central device in the linear theory is the auxiliary sequence $(c_{f,n})$, defined by
$$
c_{f,0}=0,\qquad c_{f,1}=1,\qquad
c_{f,n}=\min\{k<n:mk+c_{f,k}\ge n\}\quad(n\ge 2).
$$
It is well-defined, ascending, and satisfies
$$
c_{f,n+1}\in\{c_{f,n},c_{f,n}+1\}.
$$
In $J_\infty(f)$ one then has
$$
d^-(v_n)=n-c_{f,n},\qquad
d^+(v_n)=((m-1)n+c)+c_{f,n},
$$
together with
$$
d^+(v_n)+d^-(v_n)=f(n)=mn+c,
$$
and the slow-growth rule
$$
d^-(v_{n+1})\in\{d^-(v_n),d^-(v_n)+1\}.
$$
For a fixed head $v_j$, the minimal tail index is exactly
$$
k_j=c_{f,j},
$$
and
$$
(v_i,v_j)\in A(J_\infty(f)) \iff c_{f,j}\le i<j.
$$
Thus every in-neighborhood is a contiguous index interval [1506.06538].

| Quantity | Formula |
|---|---|
| Total degree in $J_\infty(f)$ | $d(v_k)=mk+c$ |
| Auxiliary sequence | $c_{f,n}=\min\{k<n:mk+c_{f,k}\ge n\}$ |
| In-degree | $d^-(v_n)=n-c_{f,n}$ |
| Out-degree | $d^+(v_n)=((m-1)n+c)+c_{f,n}$ |
| Minimal tail to $v_j$ | $k_j=c_{f,j}$ |

In the finite truncation $J_n(f)$, the in-degree of $v_i$ 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
$$
d_n^+(v_i)=\min\{f(i)-d^-(v_i),\,n-i\},
$$
so
$$
d_n(v_i)=d^-(v_i)+d_n^+(v_i)\le f(i).
$$
Equality holds whenever the infinite out-neighborhood of $v_i$ fits entirely inside $\{i+1,\dots,n\}$, that is, whenever
$$
(f(i)+i)-d^-(v_i)\le n.
$$
The order-$a$ theory has the analogous sequence $c_{a,n}$ and degree identity
$$
d^+(v_n)=(a-1)n+c_{a,n},
$$
with $d^-(v_n)=n-c_{a,n}$ [1506.06538] [1404.1714].

## 3. Finite structure, adjacency matrix, and extremal vertices

Because every arc goes from lower to higher index, both $J_\infty(f)$ and each $J_n(f)$ are acyclic directed graphs with natural topological ordering $v_1,v_2,\dots$. In matrix form, when rows and columns are indexed by vertex order, the adjacency matrix is upper triangular, and each column $j$ has a contiguous block of $1$’s from row $c_{f,j}$ to row $j-1$. This columnwise interval structure is one of the characteristic signatures of linear Jaco graphs [1506.06538] [1404.0484].

The underlying undirected graph $J_n^*(f)$ is connected for $m\ge 1$. The paper also isolates disconnected limiting families: if $m=0$ and $c>0$, then
$$
J_n(k)=\bigsqcup_{\lfloor n/(k+1)\rfloor \text{ copies}} K_{k+1}\ \bigsqcup\ K_{n-(k+1)\lfloor n/(k+1)\rfloor},
$$
while $J_n(0)$ is the null graph. For finite linear Jaco graphs, the minimum degree satisfies
$$
0\le \delta(J_n(f))\le f(1)=m+c,
$$
and consecutive degree differences obey
$$
|d(v_i)-d(v_{i-1})|\le m.
$$
In the order-$1$ case, the underlying undirected graph also contains the monotone Hamiltonian path
$$
v_1\to v_2\to \cdots \to v_n,
$$
since $(v_i,v_{i+1})$ is always present [1506.06538] [1404.0484].

Vertices attaining $\Delta(J_n(f))$ are the Jaconian vertices, denoted $\mathbb{J}(J_n(f))$; the least-indexed such vertex is the prime Jaconian vertex. If $v_i$ is prime Jaconian, then the Hope subgraph $\mathbb{H}(J_n(f))$ is the complete subgraph induced by $v_{i+1},\dots,v_n$. A useful rigidity statement is that if the prime Jaconian vertex $v_i$ achieves $d(v_i)=f(i)$ in $J_n^*(f)$, then all vertices $v_m$ with $m\le i$ also satisfy $d(v_m)=f(m)$, and $v_{i+1},\dots,v_n$ form a clique in the underlying graph. In the order-$1$ theory, the Jaconian set has cardinality at most $3$ [1506.06538] [1404.0484].

The literature also gives criteria for identifying the prime Jaconian vertex. For example, if $i$ is the smallest index with $d(v_i)=f(i)$ and $(v_i,v_n)$ exists, then $v_i$ is prime Jaconian. In the order-$1$ setting, $v_i$ is prime Jaconian in $J_n(1)$ if and only if $d(v_\ell)\le d(v_i)=i$ for all $\ell\in\{i+1,\dots,n\}$ [1506.06538] [1404.0484].

## 4. Enumeration, edge counts, and arithmetic specializations

Arc counting is explicit only in special regimes. For $f(x)=mx$ and $\ell\le m+1$,
$$
\varepsilon(J_\ell(mx))=\frac12 \ell(\ell-1),
$$
because $J_\ell(mx)$ is complete. More generally, if $\Delta(J_n(mx))=k$, then
$$
\varepsilon(J_n(mx))
=
\frac12 (n-k)(n-k-1)+\sum_{i=1}^k d^+(v_i),
$$
since the Hope subgraph contributes $\frac12 (n-k)(n-k-1)$ arcs and the remaining contribution comes from the first $k$ vertices. For $c=0$, the recursive update
$$
\varepsilon(J_{n+1}(mx))=
\begin{cases}
\varepsilon(J_n(mx))-i+n,& d(v_i)=mi,\\
\varepsilon(J_n(mx))-i+(n+1),& d(v_i)<mi,
\end{cases}
$$
holds when $v_i$ is the prime Jaconian vertex of $J_n(mx)$. A further special closed form is given at
$$
N=m^2+m(c+1)+2,
$$
where
$$
\varepsilon(J_N(f))
=
\varepsilon(K_{m+c+1})
+\varepsilon(K_{m^2+m(c-1)-c+2})
+\frac12 m(m+c)(m+c-1).
$$
The same paper states that finding a closed formula for $\varepsilon(J_n(f))$ for general $m,c,n$ is the chief open problem of the linear theory [1506.06538].

In the order-$1$ case, the edge-count note gives three exact formulas for $e(J_n(1))$. The basic recursion is
$$
e(J_n(1))=e(J_{n-1}(1))+d^-(v_n),\qquad e(J_1(1))=0.
$$
Equivalently,
$$
e(J_n(1))=\sum_{i=1}^n d^-(v_i)
=\frac{n(n+1)}{2}-\sum_{i=1}^n d^+(v_i),
$$
and for $n\ge 2$,
$$
e(J_n(1))=\left(\frac{n(n+1)}2-1\right)-\sum_{i=2}^n d^+(v_i).
$$
A third formula decomposes $e(J_n(1))$ at a Jaconian index $m$ when $n=m+d^+(v_m)$ [1409.0656].

These counting formulas are tied to Fibonacci arithmetic through Bettina’s Theorem. If
$$
n=f_{i_1}+f_{i_2}+\cdots+f_{i_r}
$$
is the Zeckendorf representation of $n$, then in $J_\infty(1)$
$$
d^+(v_n)=f_{i_1-1}+f_{i_2-1}+\cdots+f_{i_r-1}.
$$
The same phenomenon appears in the general order-$a$ theory as a Lucassian–Zeckendorf representation involving the generalized Lucas sequence $U(a,-1)$; there
$$
d^+(v_n)=(a-1)n+\sum_i a_iU_{i-1}+T(n).
$$
For $n=15$, the order-$1$ formulas give $e(J_{15}(1))=44$ [1404.0484] [1404.1714] [1409.0656].

## 5. Order-$1$ invariants, transforms, and associated graph families

The special case $J_n(1)$ supports a large invariant theory on the underlying undirected graph. The independence number is given recursively by the set
$$
\mathbb{I}=\{v_{i,j}\mid v_1=v_{1,1}\in\mathbb{I}\text{ and }v_i=v_{i,j}=v_{(d^+(v_{m,(j-1)})+m+1)}\},
$$
and $\alpha(J_n(1))=|\mathbb{I}|$. The same paper proves
$$
\chi(J_n(1))=
\begin{cases}
(n-i)+1,& \text{if and only if the edge }v_iv_n\text{ exists},\\
n-i,& \text{otherwise,}
\end{cases}
$$
where $v_i$ is the prime Jaconian vertex. It also gives $\beta(J_n(1))=n-\alpha(J_n(1))$, states that $b(J_n(1))=1$ for $n\ge 2$, and proves
$$
0\le m(J_n(1))\le 3
$$
for the murtage number [1410.8328].

Several later constructions retain the canonical Jaco orientation. For the Mycielski Jaco graph, the brush-number result is
$$
b_r(\mu(J_n(1)))=2\sum_{i=1}^n d^+_{J_n(1)}(v_i)\qquad (n\ge 2),
$$
and the terminal brush positions after an optimal cleaning sequence form a brush centre of $J_n(1)$. For the competition graph, the characterization for $n\ge 5$ is
$$
C(J_n(1))
=
\langle V^*\rangle
-
\{v_iv_{m_i}\mid m_i=i+d^+_{J_n(1)}(v_i),\ 3\le i\le n-2\}
\cup \{v_1,v_2,v_n\},
$$
where $V^*=\{v_i\mid 3\le i\le n-1\}$. The same paper introduces the grog number and gives the recursion
$$
g(J_{n+1}(1))=g(J_n(1))+(2i-n)+1
$$
when $v_i$ is a Jaconian vertex of $J_n(1)$ [1501.01381] [1502.01824].

Triangle structure is encoded by the primitive hole number on the underlying graph $J_n^*(1)$. If $v_i$ is the prime Jaconian vertex of $J_n(1)$, then
$$
h(J^*_{n+1}(1))=h(J^*_n(1))+\binom{n-i}{2},
$$
and for $n\ge 5$,
$$
h(J^*_n(1))
=
\binom{n-i}{3}
+
\sum_{\substack{j\le i\\ d'_{J^*_n(1)}(v_j)\ge 2}}
(d'_{J^*_n(1)}(v_j)-1).
$$
The irregularity paper studies the underlying graphs $J_n^*(x)$ via
$$
irr_t(G)=\frac12\sum_{u,v\in V(G)}|d(u)-d(v)|
$$
and
$$
firr_t(G)=\sum_{1\le i<j\le n}|F_{d(v_i)}-F_{d(v_j)}|,
$$
with recursive updates based on the prime Jaconian vertex [1503.04526] [1406.6168].

## 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 $(u_i,u_j)\notin A(G)$ when $i>j$ and at least one source vertex. In this framework, linear Jaco graphs arise from the sequence $s_1=\{i\}$, so
$$
(u_i,u_j)\in A(J_\infty(s_1)) \iff 2i\ge j,
$$
and $J_n(s_1)$ is the induced finite truncation. The graph has unique source $u_1$ and primary sink $u_n$. 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 $J_8(s_1)$,
$$
b^\bullet(J_8(s_1))=7,
$$
with
$$
B_7(J_8(s_1))
=
\{(u_3,u_4),(u_5,u_6),(u_5,u_7),(u_5,u_8),(u_6,u_7),(u_6,u_8),(u_7,u_8)\},
$$
and total black energy
$$
\left(\frac78\xi+7mc^2\right)
$$
joules. The same paper emphasizes that although $J_n(s_1)$ is a DAG, its underlying undirected graph may contain cycles, so black energy dissipation can occur [1607.00472].

A distinct recent experimental program studies the undirected graphs $J_n(x)$ and reports verified Beatty-type formulas
$$
t_1(v_i)=\left\lfloor\frac{i+1}{\varphi^2}\right\rfloor,
\qquad
\varepsilon(J_n(x))=\sum_{i=1}^n \left\lfloor\frac{i+1}{\varphi^2}\right\rfloor,
$$
together with conjectures such as
$$
\Delta(J_n(x))=\left\lfloor\frac{n+2}{\varphi}\right\rfloor-1.
$$
Further conjectures place unique $\Delta$-values, orders with $|\Delta\text{-set}|=1,2,3$, and canonical $\gamma$-set indices into Wythoff and Beatty families, while the paper marks them explicitly as conjectural rather than proved [2507.16500].

Open directions recur throughout the literature. The linear-function paper identifies the search for a closed formula for $\varepsilon(J_n(f))$ for general $m,c,n$ as the chief unresolved counting problem and proposes extensions to polynomial Jaco graphs $J_n(f)$ with
$$
f(x)=\sum_{i=1}^t a_ix^i+c.
$$
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 [1506.06538] [1607.00472] [2507.16500].

Source: https://www.emergentmind.com/topics/linear-jaco-graphs