Maximizing the sum of the squares of the degrees of a graph

Let <f>G=(V,E)</f> be a simple graph with <f>n</f> vertices, <f>e</f> edges, and vertex degrees <f>d1,d2,…,dn</f>. Also, let <f>d1</f>, <f>dn</f> be, respectively, the highest degree and the lowest degree of <f>G</f> and <f>mi</f> be the average of the degrees of the vertices adjacent to vertex <f>vi∈V</f>. It is proved thatwith equality if and only if <f>G</f> is an <f>Sn</f> graph <f>(K1,n-1⊆Sn⊆Kn)</f> or a complete graph of order <f>n-1</f> with one isolated vertex. Using the above result we establish the following upper bound for the sum of the squares of the degrees of a graph <f>G</f>:with equality if and only if <f>G</f> is a star graph or a regular graph or a complete graph <f>Kd1+1</f> with <f>n-d1-1</f> isolated vertices. A comparison is made to another upper bound on <f>∑i=1n di2</f>, due to de Caen (Discrete Math. 185 (1998) 245). We also present several upper bounds for <f>∑i=1n di2</f> and determine the extremal graphs which achieve the bounds. [Copyright &y& Elsevier]