E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete.

Let P be a set with p elements and $\{E_1,E_2,\dots ,E_q\}$ be a collection of subsets of P with ${\bigcup }_{i=1}^q{E}_i=P$. We derive an expression for the maximum value of the difference $\left({\sum }_{j=1}^k|E_{i_j}|-\left|{\bigcup }_{i=1}^k{E}_{i_j}\right|\right)$ for $2⩽k⩽q$, where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges $\{e_1,e_2,\dots ,e_q\}$, where the set $E_i$ is the set of vertices on the edge $e_i,1⩽i⩽q$. Combining the above two results, we characterize a complete semigraph.

E. Sampathkumar 通过允许边具有两个以上的顶点，将图推广为半图。与图的情况一样，完全半图是每两个顶点彼此相邻的半图。在这篇文章中，我们概括了高斯在 1796 年提到的关于三角数的问题，并表明它是半图何时完成的决定因素。

设P是一个包含p 个元素的集合，并且$\{{乙}_1,{乙}_2,\dots ,{乙}_q\}$是的子集的集合P与${\bigcup }_{\mathrm{一世}=1}^q{乙}_{\mathrm{一世}}=磷$. 我们推导出差异的最大值的表达式$\left({\sum }_{j=1}^{克}|{乙}_{{\mathrm{一世}}_j}|——\left|{\bigcup }_{\mathrm{一世}=1}^{克}{乙}_{{\mathrm{一世}}_j}\right|\right)$ 为了 $2⩽克⩽q$，其中集合中的每两个集合最多可以有一个共同元素。我们证明这个结果有助于回答在顶点集P上是否存在一个有边的半图的问题$\{{\mathrm{电子}}_1,{\mathrm{电子}}_2,\dots ,{\mathrm{电子}}_q\}$，其中集合 ${乙}_{\mathrm{一世}}$ 是边上的顶点集 ${\mathrm{电子}}_{\mathrm{一世}},1⩽\mathrm{一世}⩽q$. 结合以上两个结果，我们刻画了一个完整的半图。