On a characteristic of a conditional distribution of configuration graph

Abstract We consider configuration graphs with N vertices. The vertex degrees are independent identically distributed random variables and for any vertex of the graph the distribution of its degree η satisfies the following condition: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi mathvariant="bold">P</m:mi> </m:mrow> <m:mo fence="false" stretchy="false">{</m:mo> <m:mi>η</m:mi> <m:mo>=</m:mo> <m:mi>k</m:mi> <m:mo fence="false" stretchy="false">}</m:mo> <m:mo>∼</m:mo> <m:mfrac> <m:mi>d</m:mi> <m:mrow> <m:msup> <m:mi>k</m:mi> <m:mi>g</m:mi> </m:msup> <m:mrow> <m:msup> <m:mo form="prefix">ln</m:mo> <m:mi>h</m:mi> </m:msup> </m:mrow> <m:mi>k</m:mi> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>k</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo>,</m:mo> </m:math> $$\mathbf{P}\{\eta=k\} \sim \frac{d}{k^g \ln ^h k}, \quad k \rightarrow \infty,$$ where d &gt; 0, h ⩾ 0, 2 &lt; g &lt; 3. We obtain the limit distributions of the maximal degree of vertices in the configuration graph as N , n → ∞and n / N (3 g −4)/(2 g −2) → ∞under the conditions that the sum of vertex degrees is n .