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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2102.00105 |
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| _version_ | 1866916237896843264 |
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| author | Koolen, Jack H. Lee, Jae-Ho Tan, Ying-Ying |
| author_facet | Koolen, Jack H. Lee, Jae-Ho Tan, Ying-Ying |
| contents | Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $Γ$. 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 $Γ$ with respect to $x$. The Terwilliger algebra $T(x)$ of $Γ$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), E^*_1(x), \ldots, E^*_D(x)$. Let $V$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $Γ$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, there exists a $\mathbb{C}$-vector space isomorphism $ρ:V\to V$ such that $(ρA - A ρ)V=0$ and $(ρE^*_i(x) - E^*_i(y)ρ)V=0$ for all $0\leq i \leq D$. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. For $D=2$, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For $D = 3$, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For $D=4$, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2102_00105 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Remarks on pseudo-vertex-transitive graphs with small diameter Koolen, Jack H. Lee, Jae-Ho Tan, Ying-Ying Combinatorics 05E30 Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $Γ$. 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 $Γ$ with respect to $x$. The Terwilliger algebra $T(x)$ of $Γ$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), E^*_1(x), \ldots, E^*_D(x)$. Let $V$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $Γ$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, there exists a $\mathbb{C}$-vector space isomorphism $ρ:V\to V$ such that $(ρA - A ρ)V=0$ and $(ρE^*_i(x) - E^*_i(y)ρ)V=0$ for all $0\leq i \leq D$. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. For $D=2$, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For $D = 3$, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For $D=4$, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive. |
| title | Remarks on pseudo-vertex-transitive graphs with small diameter |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2102.00105 |