Geodesic
Some simple groups which are determined by their character degree graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 1290-1299

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Let $G$ be a finite group, and let $\rho(G)$ be the set of prime divisors of the irreducible character degrees of $G$. The character degree graph of $G$, denoted by $\Delta(G)$, is a graph with vertex set $\rho(G)$ and two vertices $a$ and $b$ are adjacent in $\Delta(G)$, if $ab$ divides some irreducible character degree of $G$. In this paper, we are going to show that some simple groups are uniquely determined by their orders and character degree graphs. As a consequence of this paper, we conclude that $M_{12}$ is not determined uniquely by its order and its character degree graph.
Keywords: character degree, minimal normal subgroup, Sylow subgroup. 
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     author = {S. Heydari and N. Ahanjideh},
     title = {Some simple groups which are determined by their character degree graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1290--1299},
     publisher = {mathdoc},
     volume = {13},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2016_13_a39/}
}
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S. Heydari; N. Ahanjideh. Some simple groups which are determined by their character degree graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 1290-1299. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a39/