A new characterization of some simple groups by order and degree pattern of solvable graph
B Akbari, N Iiyori… - Hokkaido Mathematical …, 2016 - projecteuclid.org
B Akbari, N Iiyori, AR Moghaddamfar
Hokkaido Mathematical Journal, 2016•projecteuclid.orgThe solvable graph of a finite group $ G $, denoted by ${\Gamma} _ {\rm s}(G) $, is a simple
graph whose vertices are the prime divisors of $| G| $ and two distinct primes $ p $ and $ q $
are joined by an edge if and only if there exists a solvable subgroup of $ G $ such that its
order is divisible by $ pq $. Let $ p_1< p_2<\cdots< p_k $ be all prime divisors of $| G| $ and
let ${\rm D} _ {\rm s}(G)=(d_ {\rm s}(p_1), d_ {\rm s}(p_2),\ldots, d_ {\rm s}(p_k)) $, where $ d_
{\rm s}(p) $ signifies the degree of the vertex $ p $ in ${\Gamma} _ {\rm s}(G) $. We will …
graph whose vertices are the prime divisors of $| G| $ and two distinct primes $ p $ and $ q $
are joined by an edge if and only if there exists a solvable subgroup of $ G $ such that its
order is divisible by $ pq $. Let $ p_1< p_2<\cdots< p_k $ be all prime divisors of $| G| $ and
let ${\rm D} _ {\rm s}(G)=(d_ {\rm s}(p_1), d_ {\rm s}(p_2),\ldots, d_ {\rm s}(p_k)) $, where $ d_
{\rm s}(p) $ signifies the degree of the vertex $ p $ in ${\Gamma} _ {\rm s}(G) $. We will …
The solvable graph of a finite group , denoted by , is a simple graph whose vertices are the prime divisors of and two distinct primes and are joined by an edge if and only if there exists a solvable subgroup of such that its order is divisible by . Let be all prime divisors of and let , where signifies the degree of the vertex in . We will simply call the degree pattern of solvable graph of . In this paper, we determine the structure of any finite group (up to isomorphism) for which is star or bipartite. It is also shown that the sporadic simple groups and some of projective special linear groups are characterized via order and degree pattern of solvable graph.
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