The ratio of number of edges to the sum of vertices and edges of a graph. The ratio of number of edges to the number of vertices of a graph is called its beta index (it is a measure of the graph  connectivity).  Now, i want to use a term which is the ratio of number of edges to the sum of vertices and edges of a graph (is this ratio has a name already?). As per my observations, the numerical value of this ratio of a graph $G$ is always less than or equal to its subgraphs. My questions are:

What is call this ratio?


Is my observation of the numerical value of this ratio of a graph $G$ being always less than or equal to its subgraphs correct?

Edit: Discrete graph, empty graph and null graph are excluded here.
 A: Nope, this does not always hold. We can find graphs where your ratio is arbitrarily close to $1$: for example, a complete graph with $n$ vertices has a ratio of
$$
   \frac{\binom n2}{n + \binom n2} = \frac{n-1}{n+1} = 1 - \frac2{n+1}.
$$
On the other hand, you can take a subgraph consisting of, for example, two adjacent vertices and the edge between them. Here, the ratio is just $\frac{1}{2+1} = \frac13$.

Your ratio does not have a name, but it's a function of your "beta index": if a graph has $v$ vertices, $e$ edges, and beta index $\beta = \frac ev$, then the ratio $\frac{e}{v+e}$ is equal to $\frac{\beta}{\beta+1}$. It can also be found in terms of a more well-known quantity, the average degree: if a graph has average degree $d$, then it has beta index $\frac12d$, so your ratio is $\frac{d}{d+2}$.
In particular, it's an increasing function of both values, so if you take any graph $G$ and a subgraph $H$ with a smaller average degree, then the same relation will hold between their ratios: we'll have $\frac{e(G)}{v(G)+e(G)} > \frac{e(H)}{v(H) + e(H)}$.
