The topic of these problems is nodes, branches, paths and loops. And the problem itself is to determine the number of nodes, branches, and loops in the circuit that is shown. Before we do that we need to go over what the definitions for node, branch, path and loop are. A node is point in which two or more elements in a circuit have a common connection. A branch is a single path in a network composed of one circuit element and a node at each end of that circuit element. A path is a route through a network starting at one node, going to the next node, and then subsequent nodes without encountering any node more than one time. And the loop, a loop is a path of starts and ends at the same node. So in this problem we're looking for the number of nodes, we're looking for the number of branches, and a number of loops. So, let's solve this problem. The first thing is the number of nodes. How many nodes do we have? Again, a node is a point that which two or more elements have a common connection. So if we start at the bottom, we see that there's a node at the bottom of the circuit that connects the voltage source, it connects R2 and it also connects R4. If we look at the top of the circuit we have some nodes as well. On the left hand side of the circuit there is a node that connects R1 with V, with a voltage source V. In the center of the circuit we have a node in the center at the top that connects R1,R2, and R3. We also have a node over on the right hand side of the circuit that connects R3 and R4. So ultimately, the number of nodes that we have, Is equal to 4, we have four nodes in the circuit. The next thing we want to calculate is, the number of branches. So if we look at the number of branches, again a branch is a single path in a network composed of one circuit element and a node at each end. So we have branches associated with the voltage source, that's a branch. We have a branch for R1, a branch for R2, a branch for R3 and a branch for R4. So the number of branches that we have in the circuit is equal to 5. The next thing we want to determine is the number of loops. A loops is a path that starts and ends at the same node. So, it could start at the bottom node, and we can then go around various paths to determine how man loops we have. So we know we have a loop around the left hand side of the circuit. There's a loop that goes around and connects in that fashion comes back to the beginning like that. We also have a loop on the right hand side of the circuit that starts at the bottom goes up around and comes back to the bottom of the circuit, that's two loops, and then we have a third loop. A third loop goes around the outside of the circuit. Now so it covers all the elements around the outside of the circuit starts at the bottom and ends at the bottom. That's one third loop. So the number of loops in this circuit is three.