In a directed graph represented by an Adjacency Matrix A, where A[i][j] = 1 indicates an edge from vertex i to vertex j, how is the in-degree of vertex j computed?
An edge from i to j contributes to A[i][j] = 1. Therefore, incoming edges to vertex j correspond to 1s in column j. In-degree of $j = sum of elements$ in column j.
Which condition is necessary and sufficient for an undirected graph to be bipartite?
A graph is bipartite if and only if it does not contain any odd-length cycle (Konig's Theorem).
According to Euler's formula for connected planar graphs, if V is the number of vertices, E is the number of edges, and F is the number of faces, which equation holds true?
Euler's Formula for connected planar graphs states that V - E + $F = 2$.
What is the maximum number of edges in a bipartite graph with 10 vertices?
A bipartite graph $K_{a,b}$ with a + $b = 10 vertices has maximum edges$ when $a = 5$ and $b = 5$. Max edges = a * $b = 5 \cdot 5 = 25$.
If two simple graphs G1 and G2 are isomorphic, which of the following properties MUST be equal for both graphs?
Isomorphic graphs have a 1-to-1 structure-preserving vertex mapping, implying identical number of vertices, edges, and degree sequences.