On degree sums of a triangle-free graph

For a simple triangle-free k -chromatic graph G with k ? 2 the upper bound m ( n - f ( k - 2 ) ) on the sum Σ 2 ( G ) = ? x ? V ( G ) d 2 ( x ) of the squares of the degrees of G is proved, where n , m , and f ( l ) are the order of G , the size of G , and the minimum order of a triangle-free l -chromatic graph, respectively. Consequences of this bound are discussed.Moreover, we generalize the upper bound on Σ p ( G ) = ? x ? V ( G ) d p ( x ) for p = 2 to p ? 3 .

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