Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remarks on pseudo-vertex-transitive graphs with small diameter   Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set
$X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a
vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection
matrix to the $i$th subconstituent space of $\Gamma$ with respect to $x$. The
Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the semisimple
subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), \ldots,
E^*_D(x)$. Let $V=\mathbb{C}^X$ denote a $\mathbb{C}$-vector space consisting
of complex column vectors with rows indexed by $X$. We say $\Gamma$ is
pseudo-vertex-transitive whenever for any vertices $x,y \in X$, both (i) the
Terwilliger algebras $T(x)$ and $T(y)$ of $\Gamma$ are isomorphic; and (ii)
there exists a $\mathbb{C}$-vector space isomorphism $\rho:V\to V$ such that
$(\rho A - A \rho)V=0$ and $(\rho E^*_i(x) - E^*_i(y)\rho)V=0$ for all $0\leq i
\leq D$. In this paper we discuss pseudo-vertex transitivity for
distance-regular graphs with diameter $D\in \{2,3,4\}$. In the case of diameter
two, a strongly regular graph $\Gamma$ is thin, and $\Gamma$ is
pseudo-vertex-transitive if and only if every local graph of $\Gamma$ has the
same spectrum. In the case of diameter three, Taylor graphs are thin and
pseudo-vertex-transitive. In the case of diameter four, antipodal tight graphs
are thin and pseudo-vertex-transitive.

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The categories of this paper are: nlin
21
The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph