multigraph

A multigraph is a graph in which we allow more than one edge to join a pair of vertices. Two or more edges that join a pair of vertices are called parallel edges. Every graph, then, is a multigraph, but not all multigraphs are graphs.Some authors define the concept of a graph by excluding graphs with multipleedges or loops. Then if they want to consider more general graphs the multigraph is introduced. Usually, such graphs have no loops.Formally, a multigraph $G=(V,E)$ is a pair, where $E=(V^{(2)},f)$is a multiset for which $f(x,x)=0$ and $V^{(2)}$ is the set of unordered pairsof $V$.

A multigraph can be used to a matrix whose entries are nonnegative integers. To do this, suppose that $A=(a_{ij})$ is an $m\times n$matrix of nonnegative integers.Let $V=S\cup T$, where $S=\{1,\mathrm{\dots },m\}$ and$T=\{1^{\prime },\mathrm{\dots },n^{\prime }\}$ and connect vertex $i\in S$ to vertex $j^{\prime }\in T$ with $a_{ij}$edges.