Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On distance, geodesic and arc transitivity of graphs   We compare three transitivity properties of finite graphs, namely, for a
positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and
$s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive
but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there
are infinitely many geodesic transitive graphs with this property for each of
these values of $s$, and that these graphs can have arbitrarily large diameter
if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there
exists a graph of valency $p$ that is 2-geodesic transitive but not 2-arc
transitive if and only if $p\equiv 1\pmod 4$, and for each such prime there is
a unique graph with this property: it is an antipodal double cover of the
complete graph $K_{p+1}$ and is geodesic transitive with automorphism group
$PSL(2,p)\times Z_2$.

12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin