Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Resistance distance in directed cactus graphs   Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set
$V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices
$i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$
and $j$. The resistance distance (or, simply the resistance) between any two
vertices $i$ and $j$ in $V$ is defined by
$r_{ij}:=l_{ii}^{\dag}+l_{jj}^{\dag}-2l_{ij}^{\dag}$, where $l_{pq}^{\dagger}$
is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the
Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more
significant than the classical distance. One reason for this is, numerical
examples show that the resistance distance between $i$ and $j$ is always less
than or equal to the classical distance, i.e. $r_{ij} \leq d_{ij}$. However, no
proof for this inequality is known. In this paper, we show that this inequality
holds for all directed cactus graphs.

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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: gr-qc