A ratio of the number edges to the number of vertices of a graph $G$ is called Beta index of $G$; it is denoted by $\beta$. In the following figure, the number of edges and the number of vertices are $2$ and $3$, respectively. Therefore, $\beta=\frac{2}{3}$.
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Also, the adjacency matrix of the above graph is
$$
A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix} .
$$
Suppose, by presuming the fact that every adjacency matrix corresponds to an undirected graph, how can we calculate the beta index of a graph by simply observing its adjacency matrix without looking at the graph? For a moment, i thought that the Beta index as the ratio of the number of non-zero entries to the total entries of the adjacency matrix (but it proved to be wrong).
Best Answer
The number of ones in the adjacency matrix is the sum of degrees of the vertices which will be twice the number of edges. Hence number of edges is $\frac{\text{no of $1$s in adjacency matrix}}{2}$, and number of vertices is the number of rows or columns of the adjacency matrix