A finite simple graph $G$ is called a sum graph (integral sum graph) if there is a bijection $f$ from the vertices of $G$ to a set of positive integers $S$ (a set of integers $S$) such that $uv$ is an edge of $G$ if and only if $f\left(u\right)+f\left(v\right)\in S$. For a connected graph $G$, the sum number (the integral sum number) of $G$, denoted by $\sigma \left(G\right)$ ($\zeta \left(G\right)$), is the minimum number of isolated vertices that must be added to $G$ so that the resulting graph is a sum graph (an integral sum graph). The spum (the integral spum) of a graph $G$ is the minimum difference between the largest and smallest integer in any set $S$ that corresponds to a sum graph (integral sum graph) containing $G$. We investigate the spum and integral spum of several classes of graphs, including complete graphs, symmetric complete bipartite graphs, star graphs, cycles, and paths. We also give sharp lower bounds for the spum and the integral spum of connected graphs.

有限简单图 $G$如果存在双射，则称为和图（积分和图） $F$ 从的顶点 $G$ 到一组正整数 $\mathrm{小号}$ （一组整数 $\mathrm{小号}$）这样 $üv$ 是...的优势 $G$ 当且仅当 $F（ü）+F（v）\in \mathrm{小号}$。对于连接图$G$，总和数（整数总和）$G$，表示为 $\sigma （G）$ （$\zeta （G）$），是必须添加到的最小隔离顶点数 $G$因此结果图是一个和图（一个积分和图）。图的泡沫（积分泡沫）$G$ 是任何集合中最大和最小整数之间的最小差 $\mathrm{小号}$ 对应于包含以下项的和图（积分和图） $G$。我们研究几类图的泡沫和整数泡沫，包括完整图，对称完整二部图，星形图，循环和路径。我们还为连接图的spum和积分spum给出了尖锐的下界。