On spanning tree edge dependences of graphs

Yujun Yang*††footnotetext: *Corresponding author at E-mail address:
<EMAIL_ADDRESS>Can Xu,
School of Mathematics and Information Science, Yantai University, Yantai,
264005 P.R. China

###### Abstract

Let $\tau(G)$ and $\tau_{G}(e)$ be the number of spanning trees of a connected
graph $G$ and the number of spanning trees of $G$ containing edge $e$. The
ratio $d_{G}(e)=\tau_{G}(e)/\tau(G)$ is called the spanning tree edge density
of $e$, or simply density of $e$. The maximum density
$\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree
edge dependence of $G$, or simply dependence of $G$. Given a rational number
$p/q\in(0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that
$d_{G}(e)=p/q$, then we say the density $p/q$ is constructible. More
specially, if there exists a graph $G$ such that $\mbox{dep}(G)=p/q$, then we
say the dependence $p/q$ is constructible. In 2002, Ferrara, Gould, and Suffel
raised the open problem of which rational densities and dependences are
constructible. In 2016, Kahl provided constructions that show all rational
densities and dependences are constructible. Moreover, He showed that all
rational densities are constructible even if $G$ is restricted to bipartite
graphs or planar graphs. He thus conjectured that all rational dependences are
also constructible even if $G$ is restricted to bipartite graphs (Conjecture
1), or planar graphs (Conjecture 2). In this paper, by combinatorial and
electric network approach, firstly, we show that all rational dependences are
constructible via bipartite graphs, which confirms the first conjecture of
Kahl. Secondly, we show that all rational dependences are constructible for
planar multigraphs, which confirms Kahl’s second conjecture for planar
multigraphs. However, for (simple) planar graphs, we disprove the second
conjecture of Kahl by showing that the dependence of any planar graph is
larger than $\frac{1}{3}$. On the other hand, we construct a family of planar
graphs that show all rational dependences $p/q>\frac{1}{2}$ are constructible
via planar graphs.

Keywords: spanning tree; spanning tree edge density; spanning tree edge
dependence; resistance distance; random walks on graphs

##  1 Introduction

Let $G$ be a connected graph. A spanning tree of $G$ is a spanning subgraph
that is a tree. The number of spanning trees, denoted by $\tau(G)$, of a graph
$G$ is the total number of distinct spanning subgraphs of $G$ that are trees.
The number of spanning trees is an important structural graph invariant, which
has significant application in network theory, physics, chemistry and
engineering. Due to this reason, it has been widely studied for many years. It
is well known that Kirchhoff’s matrix tree theorem has established a formula
for computing the number of spanning trees for a general graph [2]. Since the
pioneering work of Cayley [3] who first determined the number of spanning
trees of complete graphs, the number of spanning trees has been computed for
various interesting families of graphs. For example, the number of spanning
trees have been computed for complete bipartite graphs [4, 5], complete
multipartite graphs [6, 7], cubic cycle $C_{N}^{3}$ and the quadruple cycle
$C_{N}^{4}$ [9], graphs formed from a complete graph by deleting branches
forming disjoint $K$-partite subgraphs [8], multi-star related graphs [10],
$K_{n}$-complements of quasi-threshold graphs [11], circulant graphs[13, 14,
15, 16, 17, 18], $K_{n}$-complements of asteroidal graphs [19], graphs with
rotational symmetry [20], $K_{n}^{m}\pm G$ graphs [12], irregular line graphs
[21] and line graphs [22, 23], self-similar fractal models [24], 2-separable
networks [25], a type of generalized Farey graphs [26], nearly complete
bipartite graphs [27], Bruhat graph of the symmetric group[28], and so on.

In this paper, we concentrate on the number of spanning trees containing a
specific edge. For $e\in E(G)$, let $\tau_{G}(e)$ denote the number of
spanning trees of $G$ which contain $e$. The ratio
$d_{G}(e)=\tau_{G}(e)/\tau(G)$ is called the spanning tree edge density [29]
of $e$, or simply density of $e$. The spanning tree edge dependence [29] of
$G$, or simply dependence of $G$, denoted by $\mbox{dep}(G)$, is defined as
$\mbox{dep}(G)=\max_{e\in G}d_{G}(e)$. The spanning tree edge density and
dependence have important applications in networks. For instance, if we
construct a network $\mathcal{N}$ by replacing each edge of $G$ with a unit
resistor, then for any edge $e=uv$, the density of $e$ is equal to the
resistance distance between $u$ and $v$ in $\mathcal{N}$. Here, the resistance
distance between any two vertices $i$ and $j$ in a connected graph $G$,
denoted by $\Omega(i,j)$, is defined as the net effective resistance between
them in $\mathcal{N}$[30].

It is obvious that $0<d_{G}(e)\leq 1$ and $d_{G}(e)=1$ if and only if $e$ is a
cut edge. Let $p/q$ be a positive rational number, $p<q$. If there exists a
graph $G$ with edge $e\in E(G)$ such that $d_{G}(e)=p/q$, we say that the
spanning tree edge density $p/q$ is constructible. Similarly, if there exists
a graph $G$ with $\mbox{dep}(G)=p/q$, we say that the spanning tree edge
dependence $p/q$ is constructible.

In 2002, Ferrara, Gould and Suffel [29] proposed the following realizability
open problems.

Problems. Which rational spanning tree edge densities are constructible? More
specifically, which rational spanning tree edge dependences are constructible?

In 2016, Nathan Kahl [1] solved the problems by ingenious construction of
graph families. By constructing necklace graphs with complete graphs (the
definition is given in Section 2), he proved that all rational spanning tree
edge densities and dependences are constructible, even if $G$ is restricted to
claw-free graphs. In addition, by constructing necklace graphs with complete
bipartite graphs, he proved that all rational spanning tree edge densities are
constructible even if $G$ is restricted to bipartite graphs. By constructing
general theta graphs (the definition is given in Section 3), he proved that
all rational spanning tree edge densities are constructible even if $G$ is
restricted to planar graphs.

Undoubtedly, it is usually very challenging to determine which edge has the
maximum density in a graph. Due to this reason, it remains unknown whether all
rational spanning tree edge dependences are constructible if $G$ is restricted
to bipartite graphs or planar graphs, although Kahl has shown that all
rational spanning tree edge densities are constructible with these graphs. So,
Kahl proposed the following conjectures.

Conjecture 1. Let $p,q$ be positive integers, $p<q$. There exists some
function $f(p,q)$ such that, if $G$ is the bipartite construction of Theorem
2.1 (see Section 2), then $t_{i}\geq f(p,q)$ for all $2\leq i\leq n$ implies
that $\mbox{dep}(G)=d_{G}(e_{1})$.

Conjecture 2. Let $p,q$ be positive integers, $p<q$. There exists a planar
graph $G$ such that $\mbox{dep}(G)=p/q$.

In the present paper, firstly, we do find such a function $f(p,q)$ that, if
$G$ is the bipartite construction of Theorem 2.1, then $t_{i}\geq f(p,q)$ for
all $2\leq i\leq n$ implies that $\mbox{dep}(G)=d_{G}(e_{1})$, which confirms
Conjecture 1. Secondly, by electrical network approach, we show that
Conjecture 2 is true for planar multigraphs. However, for (simple) planar
graphs, we disprove the second conjecture of Kahl by showing that the
dependence of any planar graph is larger than $\frac{1}{3}$. Hence all
rational dependences $p/q$ for $p/q\leq\frac{1}{3}$ are not constructible. On
the other hand, we construct a family of planar graphs that show all rational
dependences $p/q>\frac{1}{2}$ are constructible via planar graphs.

##  2 Constructing spanning tree edge dependences via bipartite graphs

In this section, we show that Conjecture 1 is true. To this end, we use the
same bipartite construction as given in [1]. So, first of all, we introduce
the construction of necklace graphs.

Informally speaking, the necklace graph is obtained from a cycle by replacing
each edge of the cycle with a graph. Precisely, let $\\{G_{i}\\}_{i=1}^{n}$ be
a sequence of graphs, and in each graph $G_{i}$, we choose an edge
$e_{i}=u_{i}v_{i}$. Then the necklace graph
$G=N(G_{1}(e_{1}),\cdots,G_{n}(e_{n}))$ is the graph obtained from
$\\{G_{i}\\}_{i=1}^{n}$ by connecting these $n$ edges in turn according to the
sequence of the graph in which they are lied, that is, identify each $v_{i}$
and $u_{i+1}$(the indices are modulo $n$). If there is no confusion, for the
sake of simplicity, we write $G=N(G_{1},\cdots,G_{n})$ instead. Let $K_{r,s}$
be the complete bipartite graph with two partite sets having $r$ and $s$
vertices, respectively. Then the bipartite construction used in this section
is the necklace graph
$G=N(K_{r_{1},s_{1}},K_{r_{2},s_{2}}\cdots,K_{r_{n},s_{n}})$ constructed from
complete bipartite graphs. Clearly, if $n$ is even, then
$G=N(K_{r_{1},s_{1}},K_{r_{2},s_{2}}\cdots,K_{r_{n},s_{n}})$ is a bipartite
graph. For example, the necklace graph $G=N(K_{1,1},K_{2,3},K_{3,3},K_{3,4})$
is shown in figure 1.

Using the bipartite construction, Kahl proved that for any positive rational
number $0<p/q<1$, the spanning tree edge density $p/q$ is constructible, as
stated in the following theorem.

######  Theorem 2.1.

[1] Let $p,\ q$ be positive integers, $p<q$, let $t_{i}$, $2\leq i\leq n$, be
positive integers such that

$\sum\limits_{i=2}\limits^{n}\frac{1}{t_{i}}=\frac{p}{q-p}.$

Let $r_{1}=s_{1}=1$ and, for all $2\leq i\leq n$, let $r_{i}=2t_{i}$ and
$s_{i}=2t_{i}-1$. Then in $G=N(K_{r_{1},s_{1}},K_{r_{2},s_{2}},\ \cdots,\
K_{r_{n},s_{n}})$, we have $d_{G}(e_{1})=p/q$.

Figure 1: The necklace graph $G=N(K_{1,1},\ K_{2,3},\ K_{3,3},\ K_{3,4})$.

To show that the spanning tree edge dependence $p/q$ is constructible in the
bipartite construction as given in Theorem 2.1, it is needed to compute
densities of all the edges in
$G=N(K_{1,1},K_{r_{2},s_{2}}\cdots,K_{r_{n},s_{n}})$. For convenience, we
divide all the edges into three groups. The first group consists of all the
edges $e_{i}$ ($i=1,\ 2,\cdots,\ n$), these edges are called key edges of $G$,
and the vertices $u_{i}$ and $v_{i}$ are the key vertices of $G$. The second
group consists of those non-key edges which are adjacent to key edges, and we
call edges in the second group type 1 edges. The third group consists of the
remaining edges, and these edges are called type 2 edges.

Actually, to single out the edge in
$G=N(K_{1,1},K_{r_{2},s_{2}}\cdots,K_{r_{n},s_{n}})$ with the maximum spanning
tree edge density, it suffices to compute $\tau_{G}(xy)$ for all the edges
$xy\in E(G)$. To compute $\tau_{G}(xy)$, the following lemma plays an
essential role. Note that the spanning thicket[32] (or spanning bitree) of a
graph is a spanning forest with exactly two components. If vertices $u$ and
$v$ lie in different components of a spanning thicket, then we say the
spanning thicket separates $u$ and $v$.

######  Lemma 2.2.

[1] Let $G=N(G_{1},\cdots,G_{n})$ be a necklace graph. Then

$\tau(G)=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\sum\limits_{i=1}\limits^{n}d_{G_{i}}(u_{i}v_{i})$

and, for any $xy\in E(G_{k})$, $1\leq k\leq n$,

$\tau_{G}(xy)=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\left[\frac{b_{G_{k}}(xy;\
u_{k},\ v_{k})}{\tau(G_{k})}+d_{G_{k}}(xy)\sum\limits_{i\neq
k}d_{G_{i}}(u_{i}v_{i})\right]$

where $b_{G_{k}}(xy;\ u_{k},v_{k})$ is the number of spanning thickets of
$G_{k}$ separating $u_{k}$, $v_{k}$ and contain the edge $xy$.

In order to compute $\tau_{G}(xy)$ in
$G=N(K_{1,1},K_{r_{2},s_{2}}\cdots,K_{r_{n},s_{n}})$ by Lemma 2.2, we still
need to two results of Ge and Dong in [27] which gives the number of spanning
trees which contain two types of subgraphs of a complete bipartite graph
$G=K_{r,s}$. For a graph $G$ and a subgraph $H$ of $G$, we use $\tau_{G}(H)$
to denote the number of spanning trees of $G$ containing $H$. The first result
concerns the number of spanning tree of $K_{r,s}$ that contains a given
subgraph tree $T$ of $K_{r,s}$.

######  Lemma 2.3.

[27] Let $T$ be any tree which is a subgraph of $K_{r,s}$. Then

$\tau_{K_{r,s}}(T)=(ms+nr-mn)r^{s-n-1}s^{r-m-1},$ (2.1)

where $m=|V(T)\bigcap X|$, $n=|V(T)\bigcap Y|$, and $(X,Y)$ is the bipartition
of $K_{r,s}$, with $|X|=r$, and $|Y|=s$.

The second result enumerates the number of spanning trees of $K_{r,s}$
containing a given matching of $K_{r,s}$.

######  Lemma 2.4.

[27] For any matching $M$ of size $l$ in $K_{r,s}$, we have

$\tau_{K_{r,s}}(M)=(r+s)^{l-1}(r+s-l)r^{s-l-1}s^{r-l-1}.$ (2.2)

Before computing densities in $G=N(K_{1,1},\ K_{r_{2},s_{2}},\ \ldots,\
K_{r_{n},s_{n}})$, the following lemma is needed.

######  Lemma 2.5.

[29, 30] Let $G$ be an edge-transitive graph with $n$ vertices and $m$ edges,
then for any edge $e$ in $G$, we have

$d_{G}(e)=\frac{n-1}{m}.$ (2.3)

Now we are ready to compute the number of spanning trees containing a non-key
edge $e$ in $G=N(K_{1,1},\ K_{r_{2},s_{2}},\ \ldots,\ K_{r_{n},s_{n}})$. For
$2\leq k\leq n$, suppose that the two partitions of $K_{r_{k},s_{k}}$ are
$X_{k}$ and $Y_{k}$, $|X_{k}|=r_{k}$ and $|Y_{k}|=s_{k}$, $u_{k}\in X_{k}$ and
$v_{k}\in Y_{k}$.

######  Theorem 2.6.

Let $G=N(K_{1,1},\ K_{r_{2},s_{2}},\ \ldots,\ K_{r_{n},s_{n}})$. If $xy\in
E(K_{r_{k},s_{k}})$ is a type 1 edge of $G$, then

$\tau_{G}(xy)=\left\\{\begin{array}[]{rr}\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{(r_{k}-1)^{2}}{r_{k}^{2}s_{k}^{2}}\right],&\\{x,y\\}\cap\\{u_{k},v_{k}\\}=\\{u_{k}\\},\\\
\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{(s_{k}-1)^{2}}{r_{k}^{2}s_{k}^{2}}\right],&\\{x,y\\}\cap\\{u_{k},v_{k}\\}=\\{v_{k}\\}.\end{array}\right.$
(2.4)

If $xy\in E(K_{r_{k},s_{k}})$ is a type 2 edge of $G$, then

$\tau_{G}(xy)=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{1}{r_{k}^{2}s_{k}^{2}}\right].$
(2.5)

###### Proof.

We first consider the case that $xy\in E(K_{r_{k},s_{k}})$ is a type 1 edge of
$G$. By Lemma 2.2, we know that

$\tau_{G}(xy)=\prod\limits_{i=1}\limits^{n}\tau(K_{r_{i},s_{i}})\left[\frac{b_{K_{r_{k},s_{k}}}(xy;\
u_{k},v_{k})}{\tau(K_{r_{k},s_{k}})}+d_{K_{r_{k},s_{k}}}(xy)\sum\limits_{i\neq
k}d_{K_{r_{i},s_{i}}}(u_{i}v_{i})\right].$ (2.6)

For the complete bipartite graph $K_{r,s}$, it has been obtained in [31] that

$\tau(K_{r,s})=r^{s-1}s^{r-1}.$ (2.7)

In addition, since $K_{r,s}$ is edge-transitive, the spanning tree edge
density of any edge $e$ of $K_{r,s}$ is

$d_{K_{r,s}}(e)=\frac{r+s-1}{rs}.$ (2.8)

Now consider $b_{K_{r_{k},s_{k}}}(xy;\ u_{k},v_{k})$. Since $xy$ is a type 1
edge, $xy$ is adjacent to the key edge $u_{k}v_{k}$. Suppose that $y=u_{k}$ is
the common end-vertex of $xy$ and $u_{k}v_{k}$. Then the path $P=xyv_{k}$ is
tree which is a subgraph of $K_{r_{k},s_{k}}$. For each spanning tree $T$ of
$K_{r_{k},s_{k}}$ containing $P$, if we delete $u_{k}v_{k}$ from $T$, then we
could obtain a spanning thicket $B$ of $K_{r_{k},s_{k}}$ separating $u_{k}$
and $v_{k}$ that contains the edge $xy$. Conversely, for each spanning thicket
$B$ of $K_{r_{k},s_{k}}$ separating $u_{k}$ and $v_{k}$ that contains the edge
$xy$, if we add the edge $u_{k}v_{k}$ to $B$, then we obtain a spanning tree
$T$ of $K_{r_{k},s_{k}}$ containing $P$. It establishes a one to one
correspondence between the set of spanning trees of $K_{r_{k},s_{k}}$
containing $P$ and the set of spanning thickets of $K_{r_{k},s_{k}}$
separating $u_{k}$ and $v_{k}$ that contains the edge $xy$. Hence

$b_{K_{r_{k},s_{k}}}(xy;\ u_{k},v_{k})=\tau_{K_{r_{k},s_{k}}}(P).$ (2.9)

Then by Lemma 2.3, we have

$\tau_{K_{r_{k},s_{k}}}(P)=(s_{k}+2r_{k}-2)r_{k}^{s_{k}-3}s_{k}^{r_{k}-2},$
(2.10)

Substituting results of Eqs. (2.7), (2.8), and (2.10) into Eq. (2.6), we
obtain

$\displaystyle\tau_{G}(xy)$
$\displaystyle=\prod\limits_{i=1}\limits^{n}\tau(K_{r_{i},s_{i}})\left[\frac{b_{K_{r_{k},s_{k}}}(xy;\
u_{k},v_{k})}{\tau(K_{r_{k},s_{k}})}+d_{K_{r_{k},s_{k}}}(xy)\sum\limits_{i\neq
k}d_{K_{r_{i},s_{i}}}(u_{i}v_{i})\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{(s_{k}+2r_{k}-2)r_{k}^{s_{k}-3}s_{k}^{r_{k}-2}}{r_{k}^{s_{k}-1}s_{k}^{r_{k}-1}}+\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i\neq
k}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{s_{k}+2r_{k}-2}{r_{k}^{2}s_{k}}+\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\left(\sum\limits_{i=1}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\right)\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}+\frac{s_{k}+2r_{k}-2}{r_{k}^{2}s_{k}}-\left(\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\right)^{2}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{(r_{k}-1)^{2}}{r_{k}^{2}s_{k}^{2}}\right].$

Hence the first equality in Eq. (2.4) is obtained. The second equality in Eq.
(2.4) could be obtained in the same way.

Now suppose that $xy\in E(K_{r_{k},s_{k}})$ is a type 2 edge of $G$. Then $xy$
and $u_{k}v_{k}$ form a matching $M$ of $K_{r_{k},s_{k}}$, where both $x_{k}$
and $y_{k}$ distinct from key vertices $u_{k},\ v_{k}$. It is not difficult to
verify that there exists a one to one correspondence between the set of
spanning trees of $K_{r_{k},s_{k}}$ containing $M$ and the set of spanning
thickets of $K_{r_{k},s_{k}}$ separating $u_{k}$ and $v_{k}$ that contains the
edge $xy$. Thus by Lemma 2.4, we have

$b_{K_{r_{k},s_{k}}}(xy;u_{k},v_{k})=\tau_{K_{r_{k},s_{k}}}(M)=(r_{k}+s_{k})(r_{k}+s_{k}-2)r_{k}^{s_{k}-3}s_{k}^{r_{k}-3}.$
(2.11)

Substituting results of Eqs. (2.7), (2.8), and (2.11) into Eq. (2.6), we
obtain

$\displaystyle\tau_{G}(xy)$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{(r_{k}+s_{k})(r_{k}+s_{k}-2)r_{k}^{s_{k}-3}s_{k}^{r_{k}-3}}{r_{k}^{s_{k}-1}s_{k}^{r_{k}-1}}+\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i\neq
k}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{(r_{k}+s_{k})(r_{k}+s_{k}-2)}{r_{k}^{2}s_{k}^{2}}+\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{(r_{k}+s_{k}-1)^{2}}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{1}{r_{k}^{2}s_{k}^{2}}\right].$

Hence Eq. (2.5) is proved. ∎

To show that all rational spanning tree dependences are constructible via
bipartite graphs, the following result is needed.

######  Lemma 2.7.

[1] Let $G=N(G_{1},\cdots,G_{n})$ be a necklace graph with key edges $e_{i}\in
E(G_{i})$, $1\leq i\leq n$. Then $d_{G}(e_{k})\leq d_{G}(e_{l})$ if and only
if $d_{G_{k}}(e_{k})\leq d_{G_{l}}(e_{l})$.

Now we are ready to give the main result of this section.

######  Theorem 2.8.

Let $p,\ q$ be positive integers, $p<q$. Let $t_{i}$, $2\leq i\leq n$, be
positive integers such that

$\sum\limits_{i=2}\limits^{n}\frac{1}{t_{i}}=\frac{p}{q-p}\quad\mbox{and}\quad
t_{i}\geq\frac{q}{p}.$

Let $r_{i}=2t_{i}$ and $s_{i}=2t_{i}-1$, for all $2\leq i\leq n$. Then in
$G=N(K_{1,1},K_{r_{2},s_{2}},\ \cdots,\ K_{r_{n},s_{n}})$, we have
$\mbox{dep}(G)=d_{G}(e_{1})=p/q$.

###### Proof.

Since $\sum\limits_{i=2}\limits^{n}\frac{1}{t_{i}}=\frac{p}{q-p}$ satisfies
the condition in Theorem 2.1, we readily have $d_{G}(e_{1})=p/q$. To get the
required result, it suffices to show that if, for all $2\leq i\leq n$,
$t_{i}\geq\frac{q}{p}$, then $\mbox{dep}(G)=d_{G}(e_{1})$.

For any key edge $e_{k}$ ($2\leq k\leq n$), since $t_{k}\geq q/p>1$, and
$t_{k}$ is positive integers, we have $t_{k}\geq 2$. Then it follows that
$r_{k}=2t_{k}\geq 4$ and $s_{k}=2t_{k}-1\geq 3$. Thus,

$d_{K_{r_{k},s_{k}}}(e_{k})=\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}<1=d_{K_{1,1}}(e_{1}).$

Hence by Lemma 2.7, we get that for $2\leq k\leq n$,
$d_{G}(e_{k})<d_{G}(e_{1})$.

Now we compare densities of $e_{1}$ and non-key edges. By Theorem 2.6, it is
easily seen that for each $2\leq k\leq n$, if $x_{1}y_{1}\in
E(K_{r_{k},s_{k}})$ is a type 1 edge of $G$ and $x_{2}y_{2}\in
E(K_{r_{k},s_{k}})$ is a type 2 edge of $G$, then

$\tau_{G}(x_{1}y_{1})<\tau_{G}(x_{2}y_{2}).$

So, in order to show $e_{1}$ has maximum density, we only need to show that
the density of $e_{1}$ is larger than that of any type 2 edge.

By Lemma 2.2, it is easily verified that

$\tau_{G}(e_{1})=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\left[\sum\limits_{i\neq
k}d_{G_{i}}(u_{i}v_{i})\right]=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\sum\limits_{i=2}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}\right].$
(2.12)

Let $xy\in E(K_{r_{k},s_{k}})$ be a type-2 edge. Then by Theorem 2.6, we have

$\displaystyle\tau_{G}(e_{1})-\tau_{G}(xy)$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\sum\limits_{i=2}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}\right]-\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\sum\limits_{i=2}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=1}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}+\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\sum\limits_{i=2}^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}\sum\limits_{i=2}\limits^{n}\frac{r_{i}+s_{i}-1}{r_{i}s_{i}}-\frac{r_{k}+s_{k}-1}{r_{k}s_{k}}+\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\sum\limits_{i=2}^{n}\frac{4t_{i}-2}{2t_{i}(2t_{i}-1)}-\frac{4t_{k}-2}{2t_{k}(2t_{k}-1)}\sum\limits_{i=2}\limits^{n}\frac{4t_{i}-2}{2t_{i}(2t_{i}-1)}-\frac{4t_{k}-2}{2t_{k}(2t_{k}-1)}+\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{p}{q-p}(1-\frac{1}{t_{k}})-\frac{1}{t_{k}}+\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle\geq\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\left[\frac{p}{q-p}(1-\frac{p}{q})-\frac{p}{q}+\frac{1}{r_{k}^{2}s_{k}^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}r_{i}^{s_{i}-1}s_{i}^{r_{i}-1}\cdot\frac{1}{r_{k}^{2}s_{k}^{2}}>0.$

Consequently, $\mbox{dep}(G)=\max_{e\in G}d_{G}(e)=d_{G}(e_{1})=p/q$. ∎

Since in Theorem 2.8, we can always insure that $n$ is even, it implies that
the necklace graph can be always chosen to be bipartite. Thus the following
result is obvious.

######  Corollary 2.9.

Let $p$, $q$ be positive integers, $p<q$. Then there exists a bipartite graph
$G$, such that $\mbox{dep}(G)=p/q$.

##  3 Constructing spanning tree edge dependences via planar graphs

###  3.1 The case of planar multigraphs

If the graph have multiple edges, then the graph is called a multigraph. In
this section, we show that all rational spanning trees edge dependences are
constructible via planar multigraphs.

In [1], Kahl defined generalized theta graph. A generalized theta graph
$\Theta(r_{1},r_{2},\ldots,r_{n})$ is a graph consisting of two distinguished
vertices $u,v$ with $n$ disjoint paths between them, of lengths (in edges) of
$r_{1},r_{2},\ldots,r_{n}$. For example, the generalized theta graph
$\Theta(1,2,3,7)$ is shown in figure 2 (left). Using the construction of
generalized theta graph, he proved that all rational spanning tree edge
densities are constructible via planar graphs.

Figure 2: The generalized theta graph $\Theta(1,2,3,7)$ (left) and its dual
graph (right)

######  Theorem 3.1.

Let $p$, $q$ be positive integers, $p<q$. Let
$G=\Theta(1,r_{2},\ldots,r_{n})$, with

$\sum_{i=2}^{n}\frac{1}{r_{i}}=\frac{q-p}{p}.$

Then $d_{G}(uv)=p/q.$

In addition, the number of spanning trees of
$G=\Theta(r_{1},r_{2},\ldots,r_{n})$ and the number of spanning trees
containing an specific edge $e$ are also determined.

######  Lemma 3.2.

[1] Let $G=\Theta(r_{1},r_{2},\ldots,r_{n})$ be a generalized theta graph.
Then

$\tau(G)=\prod\limits_{i=1}^{n}r_{i}\left(\sum_{i=1}^{n}\frac{1}{r_{i}}\right),$

and if $e_{k}$ is an edge of the $k$th path of $G$, then

$\tau_{G}(e_{k})=\prod\limits_{i=1}^{n}r_{i}\left(\sum_{i=1}^{n}\frac{1}{r_{i}}-\frac{1}{r_{k}}\sum_{i\neq
k}\frac{1}{r_{i}}\right).$

From the above lemma, it is easily verified that in
$G=\Theta(1,r_{2},\ldots,r_{n})$, $uv$ is the unique edge with minimum
density, which seems to be depressing since the dependence of $G$ has nothing
to do with the density of $uv$. Fortunately, we could construct a new graph
$G^{*}$, called the dual graph of $G$, such that $u^{*}v^{*}$, the dual edge
of $uv$ in $G^{*}$, has maximum density in $G^{*}$.

Now we introduce the dual graph of a planar graph. Let $G$ be a planar graph
and we draw $G$ in the plane in such a way that no two edges intersect-except
at a vertex to which they are both incident. In such a drawing, $G$ is a plane
graph. The dual graph of the plane graph $G$, denoted by $G^{*}$, is a plane
graph whose vertices correspond to the faces of $G$ and the edges of $G^{*}$
correspond to edges of $G$ as follows: if $e$ is an edge of $G$ with face
$f_{1}$ on one side and face $f_{2}$ on the other side, then the end vertices
of the dual edge $e^{*}$ are the vertices that represent the faces $f_{1}$,
$f_{2}$ of $G$. We can embed $G$ and $G^{*}$ simultaneously in the plane, such
that an edge $e$ of $G$ crosses the corresponding dual edge $e^{*}$ of $G^{*}$
exact once and crosses no other edges of $G^{*}$. For example, the dual graph
of the generalized theta graph $\Theta(1,2,3,7)$ is shown in figure 2 (right).

Let $e=uv$ be an edge of $G$ and let $e^{*}=u^{*}v^{*}$ be its dual edge in
$G^{*}$. Thomassen established a nice relation between $\Omega_{G}(u,v)$ and
$\Omega_{G^{*}}(u^{*},v^{*})$.

######  Proposition 3.3.

[33] Let $G$ be a planar graph and let $G^{*}$ be the geometric dual of $G$.
Let $e=uv\in E(G)$ and $e^{*}=(u^{*},v^{*})\in E(G^{*})$ be a pair of dual
edges. Then

$\Omega_{G}(u,v)+\Omega_{G^{*}}(u^{*},v^{*})=1.$ (3.1)

From Proposition 3.3, we could draw the conclusion that all rational spanning
tree edge dependences are constructible via planar multigraphs.

######  Theorem 3.4.

Let $p<q$ be positive integers, $p<q$. Let $G=\Theta(1,r_{2},\ldots,r_{n})$,
with

$\sum_{i=2}^{n}\frac{1}{r_{i}}=\frac{p}{q-p}.$

Let $G^{*}$ be the dual graph of $G$, and let $e^{*}=u^{*}v^{*}$ be the dual
edge of $e=uv$. Then $\mbox{dep}(G^{*})=d_{G^{*}}(u^{*}v^{*})=p/q.$

###### Proof.

Since the edge $uv$ is the unique edge with minimum density in $G$, it
following from Proposition 3.3 that the edge $u^{*}v^{*}$ is the unique edge
with maximum density in $G^{*}$. By Theorem 3.1, we get that if
$\sum\limits_{i=2}^{n}\frac{1}{r_{i}}=\frac{p}{q-p},$ then $d_{G}(uv)=1-p/q$.
Hence, again by Proposition 3.3 , we get that
$\mbox{dep}(G^{*})=d_{G^{*}}(u^{*}v^{*})=p/q,$ as desired. ∎

As a straightforward consequence, we get

######  Corollary 3.5.

Let $p$, $q$ be positive integers, $p<q$. Then there exists a planar
multigraph $G$, such that $\mbox{dep}(G)=p/q$.

###  3.2 The case of (simple) planar graphs

In this section, we first show that for all $p/q\leq\frac{1}{3}$, spanning
tree edge dependences $p/q$ are not constructible via planar graphs. We first
introduce the following lemma, which is also known as the famous Foster’s
first formula in electric network theory [34].

######  Lemma 3.6.

[29] Let $G$ be a connected graph on $n$ vertices. Then

$\sum_{e\in E(G)}d_{G}(e)=n-1.$ (3.2)

By Lemma 3.6, we could get the following result.

######  Theorem 3.7.

Let $G$ be a planar graph on $n$ vertices. Then

$\mbox{dep}(G)>\frac{1}{3}.$ (3.3)

###### Proof.

Since $G$ is a planar graph, $|E(G)|\leq 3n-6$. Hence

$\mbox{dep}(G)\leq\frac{n-1}{3n-6}>\frac{1}{3}.$

∎

Theorem 3.7 implies that Conjecture 2 fails for $p/q\leq\frac{1}{3}$. That is,
all rational dependences $p/q\leq\frac{1}{3}$ are not constructible via planar
graphs.

In the rest of this section, we show that for $p/q>\frac{1}{2}$, the spanning
tree edge dependence $p/q$ is constructible via planar graphs. To this end, we
construct a new family of planar graphs. Let
$G=N(H_{r_{1}},H_{r_{2}},\cdots,H_{r_{n}})$ be a necklace graph, where
$H_{r_{i}}$ is a graph consisting of two key vertices $u_{i}$, $v_{i}$ with
one key edge $e_{i}=u_{i}v_{i}$ and $r_{i}$ disjoint paths of length 2
connecting $u_{i}$ and $v_{i}$. In particular, if $r_{i}=0$, then $H_{0}$ is
the graph only consists of the edge $u_{i}v_{i}$. Clearly, the graph
$G=N(H_{r_{1}},H_{r_{2}},\cdots,H_{r_{n}})$ is planar. For example, the graph
$G=N(H_{0},H_{4},H_{1},H_{3},H_{2},H_{3})$ is shown in Figure 3.

Figure 3: The planar graph $G=N(H_{0},H_{4},H_{1},H_{3},H_{2},H_{3})$.

Now we show that for any rational number $0<p/q<1$, the density $p/q$ is
constructible from $G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$.

######  Theorem 3.8.

Let $p,\ q$ be positive integers, $p<q$. Let
$G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$ such that

$\sum\limits_{i=2}\limits^{n}\frac{2}{r_{i}+2}=\frac{p}{q-p}.$

Then

$\mbox{d}_{G}(e_{1})=\frac{p}{q}.$

###### Proof.

We first compute $\tau(G)$. For each $H_{r_{i}}$, $1\leq i\leq n$, it is not
difficult to verify that

$\tau(H_{r_{i}})=\tau(H_{r_{i}}-e_{i})+\tau(H_{r_{i}}/e_{i})=r_{i}2^{r_{i}-1}+2^{r_{i}}=2^{r_{i}-1}(r_{i}+2),$

where $\tau(H_{r_{i}}/e_{i})$ represents the graph obtained from $H_{r_{i}}$
by contracting $e_{i}$. Hence

$d_{H_{r_{i}}}(u_{i}v_{i})=\frac{\tau_{H_{r_{i}}}(u_{i}v_{i})}{\tau(H_{r_{i}})}=\frac{\tau(H_{r_{i}}/e_{i})}{\tau(H_{r_{i}})}=\frac{2^{r_{i}}}{2^{r_{i}-1}(2+r_{i})}=\frac{2}{2+r_{i}}.$

Then by Lemma 2.2, we get

$\displaystyle\tau(G)$
$\displaystyle=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\sum\limits_{i=1}\limits^{n}d_{G_{i}}(u_{i}v_{i})=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}},$
$\displaystyle\tau_{G}(e_{1})$
$\displaystyle=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\sum\limits_{i=2}\limits^{n}d_{G_{i}}(u_{i}v_{i})=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}.$

Thus

$d_{G}(e_{1})=\frac{\tau_{G}(e_{1})}{\tau(G)}=\frac{\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}}{\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}}=\frac{\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}}{1+\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}}=\frac{\frac{p}{q-p}}{1+\frac{p}{q-p}}=\frac{p}{q}.$

∎

Then we will give the spanning tree edge number of the non-key edges in the
graph $G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$.

######  Theorem 3.9.

Let $G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$ and $r_{i}\geq 1$, for $i\geq 2$.
If $x_{k}y_{k}\in E(H_{r_{k}})$ is non-key edge of $G$, then

$\tau_{G}(x_{k}y_{k})=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}-\frac{1}{(2+r_{k})^{2}}\right].$
(3.4)

###### Proof.

To compute $\tau_{H_{r_{i}}}(x_{i}y_{i})$, according to the structure of
$H_{r_{i}}$, we can know

$\tau_{H_{r_{i}}}(x_{i}y_{i})=\tau_{H_{r_{i}}-e_{i}}(x_{i}y_{i})+\tau_{H_{r_{i}}\setminus
e_{i}}(x_{i}y_{i})=2^{r_{i}-1}+(r_{i}-1)2^{r_{i}-2}+2^{r_{i}-1}=(r_{i}+3)2^{r_{i}-2},$

and
$\tau_{H_{i}}(x_{i}y_{i};u_{i}v_{i})=\tau_{H_{r_{i}}-e_{i}}(x_{i}y_{i})=2^{r_{i}-1}$,
so
$d_{H_{r_{i}}}(x_{i}y_{i})=\frac{\tau_{H_{r_{i}}}(x_{i}y_{i})}{\tau(H_{r_{i}})}=\frac{(r_{i}+3)2^{r_{i}-2}}{2^{r_{i}-1}(2+r_{i})}=\frac{r_{i}+3}{2(2+r_{i})}$,
for any $x_{i}y_{i}\in E(H_{r_{i}})$ is non-key edge of $G$. And we known
$b_{G_{i}}(x_{i}y_{i};\ u_{i},v_{i})=\tau_{H_{i}}(x_{i}y_{i};u_{i}v_{i})$
combine with Lemma 2.2, for $x_{k}y_{k}\in E(R_{r_{k}})$

$\displaystyle\tau_{G}(x_{k}y_{k})$
$\displaystyle=\prod\limits_{i=1}\limits^{n}\tau(G_{i})\left[\frac{b_{G_{k}}(x_{k}y_{k};\
u_{k},\ v_{k})}{\tau(G_{k})}+d_{G_{k}}(x_{k}y_{k})\sum\limits_{i\neq
k}d_{G_{i}}(u_{i}v_{i})\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{2^{r_{k}-1}}{2^{r_{k}-1}(2+r_{k})}+\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i\neq
k}\limits^{n}\frac{2}{2+r_{i}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{1}{2+r_{k}}-\frac{r_{k}+3}{2(2+r_{k})}\cdot\frac{2}{2+r_{k}}+\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}-\frac{1}{(2+r_{k})^{2}}\right]$

∎

We now construct the the spanning tree edge dependence of
$G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$.

######  Theorem 3.10.

Let $p,\ q$ be positive integers, $\frac{q}{2}<p<q$. Let
$G=N(H_{0},H_{r_{2}},\cdots,H_{r_{n}})$ such that

$\sum\limits_{i=2}\limits^{n}\frac{2}{r_{i}+2}=\frac{p}{q-p}\quad\mbox{and}\quad
r_{i}\geq\frac{4q-6p}{2p-q}$

for all $2\leq i\leq n$. Then $\mbox{dep}(G)=\frac{p}{q}$.

###### Proof.

The above conditions $p<q$ and
$\sum\limits_{i=2}\limits^{n}\frac{2}{r_{i}+2}=\frac{p}{q-p}$ satisfy Theorem
3.8, we have $d_{G}(e_{1})=\frac{p}{q}$. According to Lemma 2.7, we can get
$d_{G}(e_{1})=1\geq d_{G}(e_{k})$, for any key edge $e_{k}$, with $2\leq k\leq
n$. According to Theorem 3.8 and Theorem 3.9, for any non-key edge
$x_{k}y_{k}\in E(H_{r_{k}})$,

$\displaystyle\tau_{G}(e_{1})-\tau_{G}(x_{k}y_{k})$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}-\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}-\frac{1}{(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}-\frac{r_{k}+3}{2(2+r_{k})}\sum\limits_{i=1}\limits^{n}\frac{2}{2+r_{i}}+\frac{1}{(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\left(1-\frac{r_{k}+3}{2(2+r_{k})}\right)\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}-\frac{r_{k}+3}{2(2+r_{k})}+\frac{1}{(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{r_{k}+1}{2(2+r_{k})}\sum\limits_{i=2}\limits^{n}\frac{2}{2+r_{i}}-\frac{r_{k}+3}{2(2+r_{k})}+\frac{1}{(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{r_{k}+1}{2(2+r_{k})}\frac{p}{q-p}-\frac{r_{k}+3}{2(2+r_{k})}+\frac{1}{(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\left[\frac{(r_{k}+1)(2+r_{k})}{2(2+r_{k})^{2}}\frac{p}{q-p}-\frac{(r_{k}+3)(2+r_{k})-2}{2(2+r_{k})^{2}}\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\frac{r_{k}+1}{2(2+r_{k})}\left[(2+r_{k})\frac{p}{q-p}-(r_{k}+4)\right]$
$\displaystyle=\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\frac{r_{k}+1}{2(2+r_{k})}\left(\frac{2p-q}{q-p}r_{k}-\frac{4q-6p}{q-p}\right)$
$\displaystyle\geq\prod\limits_{i=1}\limits^{n}2^{r_{i}-1}(2+r_{i})\frac{r_{k}+1}{2(2+r_{k})}\left(\frac{2p-q}{q-p}\cdot\frac{4q-6p}{2p-q}-\frac{4q-6p}{q-p}\right)$
$\displaystyle=0.$

Since for any non-key edge $x_{k}y_{k}\in E(H_{r_{k}})$, $0\leq k\leq n$, we
obtain that $\tau_{G}(e_{1})>\tau_{G}(x_{k}y_{k})$. Then
$d_{G}(e_{1})>d_{G}(x_{k}y_{k})$, so $\mbox{dep}(G)=max_{e\in
E(G)}d_{G}(e)=d_{G}(e_{1})=\frac{p}{q}$. ∎

######  Corollary 3.11.

For any rational number $\frac{1}{2}<\frac{p}{q}<1$, $p$, $q$ are positive
integers, there exists a planar graph $G$, such that
$\mbox{dep}(G)=\frac{p}{q}$.

##  4 Concluding remarks

In this paper, we show that all rational spanning tree edge dependences are
constructible for bipartite graphs, which completely solve the first
conjecture of Kahl. For the second conjecture of Kahl, we show that the
conjecture is true for planar multigraphs. However, for (simple) planar
graphs, the conjecture is not true. We show that for any rational number $p/q$
such that $p/q\leq 1/3$, the dependence $p/q$ is not constructible via planar
graphs. On the other hand, for $1/2<p/q$, we show that the dependence $p/q$ is
constructible via planar graphs. Thus, for $1/3<p/q\leq 1/2$, it still remains
an open question whether the dependence $p/q$ is constructible with planar
graphs. It deserves further discussing and studying in the future. What is
more, it is of special interest to determine the minimum rational number $p/q$
such that $p/q$ is constructible via planar graphs. So we propose the
following question.

Question. Which is the minimum rational number $0<p/q<1$ such that $p/q$ is
constructible via planar graphs?

##  5 Ackonwledgements

The support of the National Natural Science Foundation of China (through grant
no. 12171414) and the project ZR2019YQ02 by the Shandong Provincial Natural
Science Foundation, is greatly acknowledged.

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# How Pulsars Shine II: TeV emission

Andrei Gruzinov  CCPP, Physics Department, New York University, 726 Broadway,
New York, NY 10003

(October 18, 2023)

###### Abstract

Recent discovery of 20 TeV radiation from the Vela pulsar confirms
(tentatively, at the level of crude estimates) the Aristotelian
Electrodynamics picture of pulsar radiation: pulsars shine, mostly in GeV, by
annihilating colliding Poynting fluxes into curvature radiation near the light
cylinder. The observed GeV photons are the curvature radiation of
electrons/positrons with Lorentz factors $\sim 10^{7}-10^{8}$. These “super-
ultra-relativistic” electrons/positrons must also produce TeV radiation by
inverse Compton if low-energy target photons are available, as they are in the
Vela pulsar.

##  I Introduction

We first describe a scenario for the Vela pulsar’s TeV radiation, §II. This
scenario is based on Aristotelian Electrodynamics (AE, §III) as applied to
weak, Geminga-like, pulsars, §IV (weak pulsars do not produce plasma near the
light cylinder). The Vela pulsar is strong, it does produce plasma near the
light cylinder, §V. AE still applies, but it must be supplemented by a pair
production description. We will not describe the pair production explicitly.
We will combine some of the observables and the AE pulsar theory to explain
TeV (§II). 20 TeV gamma rays from the Vela pulsar HESS  confirm the AE pulsar
theory. Only tentatively, of course.

There are no new results in this paper. We merely summarize the many previous
papers to bring out the simplicity of the AE picture of pulsar radiation:

  * •

As described by Goldreich and Julian GJ  even axisymmetric pulsars spin down,
because they create plasma. The plasma supports a large-scale electromagnetic
field with an outward Poynting flux component.

  * •

Guided by the poloidal magnetic field of the neutron star, a fraction of the
Poynting flux coming from the magnetic North collides, near the light
cylinder, with a fraction of the Poynting flux coming from the magnetic South.

  * •

Goldreich and Julian GJ  postulated that the colliding Poynting fluxes will
repel and escape to infinity, without any loss of energy. In reality, that is
in AE, colliding Poynting fluxes annihilate into curvature radiation.

  * •

We know that pulsars radiate in the AE regime because the observed GeV cutoffs
are well above $\frac{mc^{2}}{\alpha}=70$MeV.

  * •

The AE pulsar theory works well for weak, Geminga-like, pulsars. For strong,
Vela-like, pulsars, soft X-rays from the star and the curvature GeV gamma rays
produce electron-positron pairs near the light cylinder. This reduces the GeV
cutoff energy and the pulsar efficiency, but curvature radiation of
electrons/positrons with Lorentz factors $\sim 10^{7}-10^{8}$ remains the
primary radiation mechanism (as tentatively confirmed by the Vela’s pulsed
TeV).

##  II TeV radiation of the Vela pulsar

We first list all the observational input and the derived data needed for the
analysis in this section and in §V. Then we describe a scenario for the Vela
pulsar’s TeV radiation.

###  II.1 Data

We need: pulsar parameters, GeV radiation, TeV radiation, high-frequency
radio, neutron star temperature. From [1-4], without error bars, as only crude
estimates will follow:

  * •

distance to Vela $d=0.29$kpc;

  * •

pulsar period $P=89$ms;

  * •

pulsar spindown rate $\dot{P}=1.25\times 10^{-13}$;

  * •

GeV energy flux $f_{\rm GeV}=9.1\times 10^{-9}\frac{\rm erg}{{\rm
cm}^{2}\cdot{\rm s}}$ with cutoff energy $E_{\rm cut}=3.0$GeV;

  * •

TeV energy flux $f_{\rm TeV}\sim 10^{-13}\frac{\rm erg}{{\rm cm}^{2}\cdot{\rm
s}}$ at $\sim 10$TeV;

  * •

Pulsar radio emission $F_{\nu}=67\mu$Jy at $\nu=344$GHz, spectral index
$\alpha=-0.93$;

  * •

$T_{\star}=0.66$MK.

Derived parameters:

  * •

GeV luminosity $L_{\rm GeV}=9.2\times 10^{34}\frac{\rm erg}{\rm s}$;

  * •

TeV luminosity $L_{\rm TeV}\sim 10^{30}\frac{\rm erg}{\rm s}$;

  * •

light cylinder radius $R=4.3\times 10^{8}$cm;

  * •

characteristic electromagnetic field in the light cylinder region $B=36$kG;

In the above: $L=4\pi d^{2}f$, $R=\frac{cP}{2\pi}$,
$cB^{2}R^{2}=\dot{E}=\frac{4\pi^{2}I\dot{P}}{P^{3}}$, $I=10^{45}{\rm
g}\cdot{\rm cm}^{2}$.

###  II.2 AE Interpretation

According to the AE picture of pulsars (§IV, §V), the $E_{\rm cut}=3.0$GeV
photons are emitted by electrons (from now on “electrons” means electrons
and/or positrons) moving along trajectories with radius of curvature $\sim R$,
at the peak of the curvature radiation power:

$E_{\rm cut}\sim 0.29\times\frac{3}{2}\gamma^{3}\frac{c\hbar}{R}.$ (1)

(Here and below, we use “$\sim$” while actually calculating to 2 digits. We
must use “$\sim$” because our assumptions, like radius of curvature $R$
instead of $2R$, are meaningful only to order of magnitude. We calculate to 2
digits to avoid error accumulation.) This gives the Lorentz factor $\gamma$
and energy $E_{e}$ of the GeV-emitting electrons

$\gamma\sim 5.3\times 10^{7},~{}~{}~{}E_{e}=27{\rm TeV}.$ (2)

###  II.3 TeV radiation

These super-ultra-relativistic electrons will scatter a target photon of
energy $E_{t}\sim\frac{mc^{2}}{\gamma}\sim 9.6\times 10^{-3}$eV into a photon
of energy $E_{\rm IC}\sim\frac{2}{3}E_{e}\sim 18$TeV with the cross section
$\sigma_{\rm IC}\sim 0.47\times\frac{8\pi}{3}r_{e}^{2}\sim 3.1\times
10^{-25}{\rm cm}^{2}$.

Photons of energy $E_{t}\sim\frac{mc^{2}}{\gamma}\sim 9.6\times 10^{-3}$eV
have frequency $\nu\sim 2.3$THz. Extrapolating from $F_{\nu}=67\mu$Jy at
$\nu=344$GHz with the spectral index $\alpha=-0.93$, we get $F_{\nu
t}=11\mu$Jy at $\nu_{t}=2.3$THz. The density of the target photons near the
light cylinder is

$n_{t}\sim\frac{F_{\nu t}}{2\pi c\hbar}\left(\frac{d}{R}\right)^{2}\sim
2.4\times 10^{12}{\rm cm}^{-3}.$ (3)

The optical depth for Compton scattering is

$\tau\sim n_{t}\sigma_{\rm IC}R\sim 3.2\times 10^{-4}.$ (4)

The number of electrons responsible for the observed gamma radiation is

$N\sim\frac{L_{\rm GeV}}{\frac{2}{3}\gamma^{4}\frac{ce^{2}}{R^{2}}}\sim
4.7\times 10^{29}.$ (5)

The Inverse Compton luminosity of these electrons is

$L_{\rm TeV,theoretical}\sim\frac{N\tau E_{\rm IC}}{R/c}\sim 3.0\times
10^{29}\frac{\rm erg}{\rm s},$ (6)

versus the observed $L_{\rm TeV}\sim 10^{30}\frac{\rm erg}{\rm s}$ –
acceptable.

The above might all be very well, but not particularly interesting unless we
explain where the (i) $N\sim 10^{30}$ electrons, (ii) with Lorentz factors
$\gamma\sim 10^{8}$, and (iii) radii of curvature $R\sim 3\times 10^{8}$cm
come from.

We will compute these three numbers:

  * •

exactly for a “pulsar device”, §III;

  * •

convincingly, although numerically, for a weak pulsar, §IV;

  * •

tentatively, very crudely, for a strong pulsar, §V.

##  III Aristotelian Electrodynamics

To the best of my knowledge, AE first appeared in Fink .

###  III.1 AE: definition and estimates

Aristotelian Electrodynamics (AE) Fink ; Gruz ; Jaco  is a simplified
description of electron/positron motion and curvature radiation in a strong
electromagnetic field. AE is valid when electrons are radiation-overdamped;
this happens when the Aristotle number (analog of the inverse Reynolds number
– the dynamical time divided by the damping time, to be explained) is large:

${\rm
Ar}\equiv\frac{e^{5/4}}{mc^{2}}B^{3/4}R^{1/2}=\left(\frac{B}{B_{c}}\right)^{3/4}\left(\frac{R}{r_{e}}\right)^{1/2}\gg
1.$ (7)

Here $r_{e}\equiv\frac{e^{2}}{mc^{2}}=2.82\times 10^{-13}$cm is the classical
electron radius, $B_{c}\equiv\frac{e}{r_{e}^{2}}=6.04\times 10^{15}$G is the
classical electron field, $B$ is the characteristic electromagnetic field, $R$
is the characteristic length scale.

In the above, one assumes that the work done by the electric field $E\sim B$
is balanced by curvature radiation:

$ceB\sim\gamma^{4}\frac{ce^{2}}{R^{2}}.$ (8)

This gives the terminal Lorentz factor

$\gamma\sim\left(\frac{BR^{2}}{e}\right)^{1/4}.$ (9)

The ratio of the dynamical time and the curvature cooling time is called Ar:

${\rm Ar}\sim\frac{R/c}{\gamma mc^{2}/\frac{\gamma^{4}ce^{2}}{R^{2}}}.$ (10)

The overdamped electrons emit curvature gamma rays of characteristic energy
(critical energy of the synchrotron spectrum)

$E_{c}\sim\frac{mc^{2}}{\alpha}\times{\rm Ar}=70{\rm MeV}\times{\rm Ar},$ (11)

$\alpha\equiv\frac{e^{2}}{c\hbar}=\frac{1}{137}$ is the fine structure
constant, because the energy of curvature photons is

$E_{c}\sim\gamma^{3}\frac{\hbar c}{R}.$ (12)

Eq.(11) immediately shows that pulsars must operate in the AE regime, as the
observed GeV cutoff energies are above about 1 GeV; assuming the observed GeV
radiation is curvature.

###  III.2 AE: precise formulation

Under the AE assumption, ${\rm Ar}\gg 1$, the velocity of positrons and
electrons is equal to the speed of light:

${\bf v}_{\pm}=\frac{{\bf E}\times{\bf B}\pm(B_{0}{\bf B}+E_{0}{\bf
E})}{B^{2}+E_{0}^{2}}~{}~{}~{}({\rm AE~{}I}).$ (13)

This is the first AE equation. It says how the charges move. Here $E_{0}$ is
the proper electric field scalar and $B_{0}$ is the proper magnetic field
pseudoscalar:

$B_{0}^{2}-E_{0}^{2}=B^{2}-E^{2},~{}B_{0}E_{0}={\bf B}\cdot{\bf
E},~{}E_{0}\geq 0.$ (14)

To derive the first AE equation, construct a future directed null 4-vector
$(1,{\bf v})$ in terms of the electromagnetic field tensor $F_{\mu\nu}=({\bf
E},{\bf B})$, say as an eigenvector of $F_{\mu\nu}$ Jaco .

The actual (very large) Lorentz factor is obtained by balancing the curvature
power ($R$ is the radius of curvature)

$q=\frac{2}{3}\gamma^{4}\frac{ce^{2}}{R^{2}},$ (15)

by the electric field power

$q=\pm ec{\bf v}_{\pm}\cdot{\bf E},$ (16)

or, using Eq.(13),

$q=ecE_{0}~{}~{}~{}({\rm AE~{}II}).$ (17)

This is the second AE equation. It says how much power the charge radiates.
All the power $q$ is emitted strictly along ${\bf v}_{\pm}$, as opening angles
$\sim\gamma^{-1}\sim 10^{-8}$ don’t matter. Knowing $\gamma$, one computes the
spectrum – synchrotron with critical energy

$E_{c}=(3/2)^{7/4}c\hbar e^{-3/4}E_{0}^{3/4}R^{1/2}~{}~{}~{}({\rm AE~{}III}).$
(18)

This is the third and last AE equation. It says with what spectrum the power
$q$ is emitted.

###  III.3 AE: exact solution – the pulsar device

The first AE equation, Eq.(13), plus Maxwell equations for the resulting
electromagnetic field have the following exact solution called the pulsar
device (in dimensionless rationalized units):

${\bf E}=(0,e^{-y},0),$ (19) ${\bf B}=(0,0,B(x,y)),$ (20)
$B(x,y)=-e^{-y}\left\\{\begin{array}[]{rl}{\rm
sign}(x),&~{}~{}~{}|x|>\frac{\pi}{2};\\\
\sin(x),&~{}~{}~{}|x|<\frac{\pi}{2}.\\\ \end{array}\right.$ (21)

There are no positrons; electrons’ number density $n$ and velocity ${\bf v}$
follow from $\nabla\cdot{\bf E}=-n$ and $\nabla\times{\bf B}=-n{\bf v}$:

$n=e^{-y},$ (22) ${\bf v}=\left\\{\begin{array}[]{rl}(-{\rm
sign}(x),0,0)~{}~{}~{}~{}~{}~{}~{}~{},&~{}~{}~{}|x|>\frac{\pi}{2};\\\
(-\sin(x),-\cos(x),0),&~{}~{}~{}|x|<\frac{\pi}{2}.\\\ \end{array}\right.$ (23)

Maxwell equations are thus satisfied. As a consequence, the continuity
equation, $\nabla\cdot(n{\bf v})=0$, is satisfied too. It only remains to be
checked that ${\bf v}={\bf v}_{-}$ as given by Eq.(13).

First compute the proper fields:

$\displaystyle|x|>\frac{\pi}{2}:~{}~{}E_{0}=0,~{}~{}B_{0}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};$
(24)
$\displaystyle|x|<\frac{\pi}{2}:~{}~{}E_{0}=e^{-y}\cos(x),~{}~{}B_{0}=0~{}~{}~{}.$
(25)

We see that $|x|>\frac{\pi}{2}$ is a force-free zone, where no electromagnetic
energy is transferred to electrons, while $|x|<\frac{\pi}{2}$ is a radiation
zone, where the electromagnetic energy is transferred to electrons and
radiated away.

In the force-free zone the electrons move along $x$, toward $x=0$, that is
toward the radiation zone. This is a pure $E\times B$ drift, at the speed of
light, as dictated by Eq.(13). The electrons carry (more precisely, are
accompanied by) a Poynting flux ${\bf S}=(-e^{-2y}{\rm sign}(x),0,0)$. The
divergence of the Poynting flux is zero, $\nabla\cdot{\bf S}=0$. The entire
Poynting flux coming from $x=\pm\infty$ reaches the radiation zone.

In the radiation zone the proper electric field $E_{0}$ appears and the
electrons are deflected toward negative $y$, still as dictated by Eq.(13):

${\bf v}_{-}=\frac{{\bf E}\times{\bf B}-E_{0}{\bf E}}{B^{2}+E_{0}^{2}}={\bf
v}.$ (26)

Here, in the radiation zone, at $|x|<\frac{\pi}{2}$, the Poynting flux is
${\bf S}=(-e^{-2y}\sin(x),0,0)$. The divergence of the Poynting flux is
negative, $\nabla\cdot{\bf S}=-e^{-2y}\cos(x)$, there is an energy sink. None
of the Poynting flux reaches the midplane $x=0$. The Poynting flux is
completely annihilated into curvature radiation.

##  IV Weak pulsar

Essentially, a weak pulsar is a pulsar device with the size equal to the light
cylinder radius and the characteristic EM field equal to the pulsar’s field at
the light cylinder.

Even the lowest spindown luminosity pulsars seen in gamma rays, like the
Geminga pulsar, have ${\rm Ar}\gtrsim 10$ near the light cylinder. AE applies
and can be used to describe pulsar radiation. This has been done in a long
series of papers Gruzon . The results are in good quantitative and/or
qualitative agreement with the Fermi Gamma-ray Space Telescope pulsar data
Fermi , but only for weak pulsars, with moderate Aristotle numbers, ${\rm
Ar}\lesssim 30$. Here we summarize the methods and results of Gruzon .

Methodologically, Gruzon  is straightforward. Take a small star – small,
compared to the light cylinder $R$. The star is a conducting rotating sphere,
with a magnetic dipole moment. Electrons and positrons are copiously deposited
near the star – copiously, compared to the Goldreich-Julian number per
rotation, $\dot{N}_{\rm GJ}\sim\frac{cBR}{e}$. Then the electrons and
positrons move and radiate as dictated by AE. The electric and magnetic fields
are given by Maxwell equations with appropriate boundary conditions at the
surface of the star, with the current density from the AE-moving electrons and
positrons.

The results are qualitatively similar to the pulsar device with the size and
field equal to the pulsar’s actual $R$ and $B$ – the light cylinder radius and
the magnetic field at the light cylinder radius. The results are approximately
independent of the size of the star – so long as the star remains within the
light cylinder radius $R$. The results are approximately independent of the
pair deposition rate – so long as the deposition rate stays above the
Goldreich-Julian rate $\dot{N}_{\rm GJ}$.

We get reasonable: lightcurves, characteristic (averaged over spin-dipole and
observation angles) efficiency of the spindown to GeV conversion,

$\epsilon\equiv\frac{L_{\rm GeV}}{\dot{E}}\sim 0.2,$ (27)

photon index

$\Gamma\approx 1,$ (28)

photon energy cutoff

$E_{\rm cut}\approx 5.2\dot{E}_{34}^{3/8}P_{\rm ms}^{-1/4}{\rm GeV}.$ (29)

The above is indeed very similar to the pulsar device. $E_{\rm cut}$ is a
numerically computed version of Eq.(18). $\epsilon\sim 0.2$ rather than
$\epsilon=1$ is because in a pulsar most of the Poynting flux escapes to
infinity without colliding. Only a fraction of the Poynting flux, guided by
the poloidal magnetic field, has to collide and annihilate near the light
cylinder.

It is clear why the vicinity of the light cylinder is (seen in the numerical
simulations of Gruzon  to be) the main region for the Poynting flux collision.
Inside the light cylinder, the poloidal magnetic field dominates, most of the
plasma simply co-rotates with the star. Outside the light cylinder, the
poloidal magnetic field becomes weaker that the radial Poynting flux. Far
outside the light cylinder, the poloidal field cannot guide the Poynting flux,
here the Poynting flux guides the poloidal field and escapes to infinity.

So the Poynting fluxes mostly collide near the light cylinder. In order to
mimic the pulsar, the size of the pulsar device must be $\sim R$, the light
cylinder radius, and the field of the pulsar device must be $\sim B$, the
pulsar field at the light cylinder.

The synchrotron photon index is $\Gamma=2/3$, but the PLEC fit to a
synchrotron spectrum that we found numerically for an axisymmetric pulsar is
$\Gamma\approx 1$, clearly due to smeared $E_{c}$. Further increasing $\Gamma$
is not a problem (and therefore not a good probe). But the observed
$\Gamma>2/3$ is significant.

One might think that a fatal flaw of our approach is the large size of the
star. Even in the best simulations we had $R_{\star}\gtrsim 0.1R$, while the
actual size is $R_{\star}\sim 0.001R$. But actually our size is exactly right.
If the plasma didn’t exist at radii $\sim 0.1R$, the electric field would have
been unscreened, $E_{0}$ would drive a test electron to high terminal Lorentz
factors, the resulting GeV curvature photons would pair produce on the stellar
soft X-rays, leading to an avalanche, similar to what happens in strong
pulsars described in §V.

##  V Strong pulsar

For a weak axisymmetric pulsar, the GeV cutoff is given by Eq.(29). For
example, for the Geminga pulsar, with $\dot{E}=3.3\times 10^{34}\frac{\rm
erg}{\rm s}$, $P=237$ms, we get $E_{\rm cut}=2.1$GeV from Eq.(29) versus the
observed $2.2\pm 0.1$ GeV. But Eq.(29) clearly fails for the Vela pulsar:
predicted $E_{\rm cut}=20$GeV versus the observed $3.0\pm 0.1$ GeV. This is
because at high Aristotle numbers pair production at the light cylinder is
inevitable, the pulsar becomes strong.

The Aristotle number, in terms of the pulsar period $P$ and the spindown power
$\dot{E}$ is

${\rm Ar}=39\times\dot{E}_{34}^{3/8}P_{\rm ms}^{-1/4},$ (30)

which is 16 for the Geminga and 147 for the Vela.

Pair production by GeV gamma-rays on soft X-ray photons from a star of
temperature $T_{\star}$ occurs if

$E_{c}\times
T_{\star}\gtrsim(mc^{2})^{2},~{}~{}~{}E_{c}\sim\frac{mc^{2}}{\alpha}\times{\rm
Ar}.$ (31)

That is for ${\rm Ar}\gtrsim{\rm Ar}_{c}$, where

${\rm Ar}_{c}\sim\frac{\alpha mc^{2}}{T_{\star}}\sim\frac{3.7{\rm
keV}}{T_{\star}}.$ (32)

For the Geminga ($T_{\star}\approx 42$eV Karg ), ${\rm Ar}_{c}\sim 88>16$. For
the Vela, $T_{\star}\approx 57$eV, ${\rm Ar}_{c}\sim 65<147$.

We must conclude that, unlike the Geminga, the Vela would pair produce if it
were in the weak pulsar regime. When the pair production threshold is
exceeded, an avalanche of pair production occurs if the mean number of pairs
produced by an electron as it crosses the light cylinder is large, $N_{\rm
pair}\gtrsim 1$. Now

$N_{\rm pair}\sim\tau_{\gamma X}N_{\rm GeV},$ (33)

where $N_{\rm GeV}$ is the number of gamma-ray photons emitted by an electron
as it crosses the light cylinder and $\tau_{\gamma X}$ is the optical depth
for pair production.

We have

$N_{\rm GeV}\sim\frac{eBR}{E_{c}},$ (34)

and

$\tau_{\gamma X}\sim\sigma_{\rm pf}Rn_{X},$ (35) $\sigma_{\rm pf}\sim
r_{e}^{2}\frac{m^{2}c^{4}}{E_{c}T_{\star}}\ln\left(\frac{E_{c}T_{\star}}{m^{2}c^{4}}\right),$
(36)
$n_{X}\sim\frac{\pi}{240}\left(\frac{R_{\star}}{R}\right)^{2}\left(\frac{T_{\star}}{c\hbar}\right)^{3},$
(37)

giving $N_{\rm pair}\sim 14$ for the Vela.

Both conditions are satisfied: we are above the threshold, ${\rm
Ar}\gtrsim{\rm Ar}_{c}$, and strongly in the avalanche regime, $N_{\rm
pair}\gg 1$. Pair production must occur near the light cylinder of the Vela
pulsar.

We know, tentatively Gruzs , that pair production reduces (i) the size of the
radiation zone, (ii) the proper electric field $E_{0}$. For example, we have
$E_{0~{}{\rm max}}=0.65B$ in AE without pair production (Fig.2 in the “Pulsar
Emission Spectrum” paper of Gruzon ), and $E_{0~{}{\rm max}}=0.20B$ in a model
AE with pair production (Fig.5 of the second paper of Gruzs ).

The pair production avalanche stabilizes as the size and the strength of the
radiation zone is reduced. But the controlled pair production never stops – it
is needed to maintain small $E_{0}$ and therefore small $E_{\rm cut}$. To get
to the observed $E_{\rm cut}\sim 3GeV$ from the weak-pulsar theoretical value
$\sim 20$GeV, we need roughly a factor of 10 “modification” of the radiation
zone. Let us check the consistency of this assumption.

The number of 3GeV-emitting electrons has been computed in §II for the Vela
pulsar: $N\sim 4.7\times 10^{29}$. The Goldreich-Julian number in the Vela is

$N_{\rm GJ}\sim\frac{BR^{2}}{e}\sim 1.4\times 10^{31},$ (38)

A factor $\sim 30$. The Vela efficiency is $\epsilon=1.3$% – a factor of $\sim
15$ below the mean weak-pulsar efficiency. Roughly a factor $\sim 10$
“modification” of the radiaton zone, as expected. This frivolous logic proves
nothing, of course; a real numerical simulation of the pair production is
highly desirable.

##  VI Conclusion

Tentatively, at the level of crude estimates, the recent discovery of 20 TeV
radiation from the Vela pulsar confirms the Aristotelian Electrodynamics
picture of the pulsar gamma-ray radiation: pulsars shine by annihilating
colliding Poynting fluxes into curvature radiation near the light cylinder. In
weak pulsars like the Geminga pulsar, this is all there is. In strong pulsars
like the Vela pulsar, pair production modifies this processes, reducing the
size and the strength of the radiation zone, but curvature radiation near the
light cylinder is still happening and is responsible for the observe GeV
photons.

In the Vela pulsar, according to the AE picture:

  * •

The observed 3 GeV photons are the curvature radiation of electrons at about
the light cylinder.

  * •

Since the curvature radius is $R\sim 4\times 10^{8}$cm, the Lorentz factor is
$\gamma\sim 5\times 10^{7}$, and the electron energy is $E\sim 30{\rm TeV}$.

  * •

Given an acceptable target photon, the $30{\rm TeV}$ electron makes a $\sim
30{\rm TeV}$ photon by Compton scattering.

## References

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  * (2) P. Goldreich, W.H. Julian, Pulsar Electrodynamics, Astrophysical Journal 157, 869 (1969)
  * (3) The Fermi-LAT collaboration, The Second Fermi Large Area Telescope Catalog of Gamma-ray Pulsars, arXiv:1305.4385 (2013)
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  * (8) T. Jacobson, Structure of Aristotelian Electrodynamics, arXiv:1504.07311 (2015)
  * (9) https://cosmo.nyu.edu/andrei/Papers/pulsars.html
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