A. A. Mironov Sov. Phys. Dokl. , vol.37 No.6 , 01.1992 , p. 278-280, language: английский ISSN 0038-5689  Аннотация We denote by $\Gamma\sb{Z,L}(c)$ and $\Gamma\sb Z(c)$ classes of unoriented multigraphs with and without loops, and we denote by $\Gamma\sb L(c)$ and $\Gamma(c)$ classes of unoriented graphs with and without loops for which the multiplicity or weight of the edges (loops) does not exceed $c\ge 0$ $[\Gamma\sb{Z,L}(1)$ and $\Gamma\sb Z(1)$ are classes of graphs with and without loops]. We call the elements of $\Gamma[c]=\Gamma\sb{Z,L}(c)\cup\Gamma\sb Z(c)\cup \Gamma\sb L(c)\cup \Gamma(c)$ generalized graphs. The order of a generalized graph $G$ from $\Gamma[c]$ is the number or sum of the weights of the edges and the loops which are incident with this vertex. The set of numbers $A=\{a\sb 1,\dots,a\sb n\}$ is called a realizable set in a generalized graph if $A$ is the set of orders of the vertices of sum $G= G(A)$ from $\Gamma[c]$, which is called a realization of $A$.\par For $n\ge 2$ we introduce $\overline R\sp n=\{A=\{a\sb 1,\dots,a\sb n\}: a\sb i\ge a\sb{i+1},\ 1\le i\le n-1;\ a\sb n\ge 0\}$ and $\overline Z\sp n=\{A\in \overline R\sp n: a\sb i\in Z,\ 1\le i\le n\}$. If $c\ge 0$, then $\overline R\sb{r,L}\sp n(c)$ and $\overline R\sb r\sp n(c)$ are subsets of sets of numbers from $\overline R\sp n$ which are realizable in weighted graphs from $\Gamma\sb L(c)$ and $\overline Z\sp n\sb r(c)$, respectively. If $c\in Z$, $c\ge 0$, then $\overline Z\sb{r,L}\sp n(c)$ and $\overline Z\sp n\sb r(c)$ are subsets of sets from $\overline Z\sp n$ which are realizable in multigraphs from $\Gamma\sb{Z,L}(c)$ and $\Gamma\sb Z(c)$.\par For $A$ from $\overline R\sp n$ or $\overline Z\sp n$ the following assertions are obvious: (a) $A$ is realizable in a generalized graph with loops. If $A\in\overline R\sp n\sb r(c)$ $[A\in \overline Z\sp n\sb r(c)]$, then $A\in \overline R\sb{r,L}\sp n(a\sb 1)$ $[A\in Z\sb{r,L}\sp n(a\sb 1)]$. (b) If $A\in \overline R\sp n\sb r(c)$ $[A\in Z\sp n\sb r(c)]$ for some $c$, then $A\in\overline R\sp n\sb r(a\sb 1)$ $[A\in\overline Z\sp n\sb r(a\sb 1)]$. We denote by $S(A)$ the sum of the numbers from {\it A. S. L. Khakimi} [Realizability of a set of integers by orders of the vertices of a graph [in Russian] (1966)] has shown that if $A\in \overline R\sp n$ [or if $A\in \overline Z\sp n$ and $S(A)= 2q$, $q\in Z$] then $A\in\overline R\sp n\sb r(c)$ $[A\in\overline Z\sp n\sb r(c)]$ for some $c$ if and only if $2a\sb 1\le S(A)$.\par For $A\in\overline R\sp n$, $c\ge 0$ and $k\in Z$, $1\le k\le n-1$, we construct two functions, whose values determine the possibility of realizability in a multigraph and in a weighted graph without and with loops.  Ключевые слова Graph theory; multigraphs; generalized graph; realizability