How many edges does a bipartite graph on m and n vertices have?a.mnb.m+nc.m+n-1d.m+n+1

To determine how many edges a bipartite graph with $m$ and $n$ vertices has, let's analyze the definition of a bipartite graph.

1. ### Break Down the Problem

A bipartite graph consists of two disjoint sets of vertices. Let's denote the sets of vertices as $U$ and $V$, where $|U| = m$ and $|V| = n$. An edge in a bipartite graph can only connect a vertex from set $U$ to a vertex in set $V$, and no edges exist between vertices within the same set.

2. ### Relevant Concepts

The maximum number of edges in a bipartite graph can be determined by the formula: $E = m \times n$ where $E$ represents the number of edges, $m$ is the number of vertices in set $U$, and $n$ is the number of vertices in set $V$.

3. ### Analysis and Detail

1. Each vertex in set $U$ can connect to every vertex in set $V$.
2. Therefore, if you have $m$ vertices in $U$ and $n$ vertices in $V$, the total number of possible edges is simply the product of the number of vertices in each set. Thus: $E = m \cdot n$

4. ### Verify and Summarize

The derived formula $E = m \cdot n$ is valid as it covers all possible edges that can be formed between the two sets in a bipartite graph, ensuring no intra-set connections.

Final Answer

The number of edges in a bipartite graph with $m$ and $n$ vertices is $mn$. Therefore, the correct answer is a. $mn$.