Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fixing number of co-noraml product of graphs   An automorphism of a graph $G$ is a bijective mapping from the vertex set of
$G$ to itself which preserves the adjacency and the non-adjacency relations of
the vertices of $G$. A fixing set $F$ of a graph $G$ is a set of those vertices
of $G$ which when assigned distinct labels removes all the automorphisms of
$G$, except the trivial one. The fixing number of a graph $G$, denoted by
$fix(G)$, is the smallest cardinality of a fixing set of $G$. The co-normal
product $G_1\ast G_2$ of two graphs $G_1$ and $G_2$, is a graph having the
vertex set $V(G_1)\times V(G_2)$ and two distinct vertices $(g_1, g_2),
(\acute{g_1}, \acute{g_2})$ are adjacent if $g_1$ is adjacent to $\acute{g_1}$
in $G_1$ or $g_2$ is adjacent to $\acute{g_2}$ in $G_2$. We define a general
co-normal product of $k\geq 2$ graphs which is a natural generalization of the
co-normal product of two graphs. In this paper, we discuss automorphisms of the
co-normal product of graphs using the automorphisms of its factors and prove
results on the cardinality of the automorphism group of the co-normal product
of graphs. We prove that $max\{fix(G_1), fix(G_2)\}\leq fix(G_1\ast G_2)$, for
any two graphs $G_1$ and $G_2$. We also compute the fixing number of the
co-normal product of some families of graphs.

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The categories of this paper are: gr-qc
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg