Minnor differences in notation used in definition of graphs

Question

One of book states

A graph G consists of two finite sets: a nonempty set V(G) of vertices and a set E(G) of edges, where each edge is associated with a set consisting of either one or two vertices called its endpoints.

Another book states

A graph is a pair of vertices $(V, \varepsilon)$ of sets, V nonempty and each element of $\varepsilon$ a set of two distinct elements of V...when we say that $G(V, \varepsilon)$ is a graph, we mean that G is a graph with vertex set V and edge set $\varepsilon$.

Concern 1: why does the first book write V(G) and E(G) instead of just V and G?

Concern 2: Combining the two definitions, does it mean a graph G is fully defined as G(G(V), G(E))?

Concern 3: wikipedia's definition of graph says edges are "2-element subsets of V" but this definition says the set can be either one or two elements from V. This makes more sense to me because an edge can be a closed loop, and in this case wouldn't it only have 1 vertex? Also the second definition says there are two distinct elements from V that make up an element in $\varepsilon$ which makes loops impossible.

Concern 4: why does the second definition use $\varepsilon$ instead of E for edges?

Answer 1

Concern 1: Sometimes we have a few graphs that we are interested in. (e.g. two graphs G and H) In those cases, it is useful to have use the $V(G)$ notation (as opposed to $V(H)$, for example). If there is only one graph in question we sometimes abbreviate to $V$ and $E$. (This "abbreviation actually happens for many parameters in graph theory if the graph in question is clear)

Concern 2: Yes, I believe so.

Concern 3: You are right: for loops we do not need two distinct elements. It is likely, however, that the author for the second book is only restricting his definitions to simple graphs (i.e. no loops or repeating edges) in which case we need the vertices to be distinct. Many elementary theorems in graph theory only focus on simple graph. Usually many introductory books put in the restriction after definitions (which can be more general) but the author may have already done it beforehand (for better or for worse).

Concern 4: I believe it is just down to notational preference. As long as the convention in your usual setting is clear (or you define terms when they may be ambiguous) neither is better or worse I suppose (I believe the $E$ notation is more common, but that may just be me.)

Answer 2

Concern 1: why does the first book write $V(G)$ and $E(G)$ instead of just $V$ and $G$?

The first book conceptualizes $V$ and $E$ as functions that accept a graph $G$ and return its vertex set $V(G)$ and its edge set $E(G)$ respectively.

Concern 2: Combining the two definitions, does it mean a graph G is fully defined as G(G(V), G(E))?

Kind of. If we're viewing $V$ and $E$ as functions, as the first writer does, and if we're also viewing a graph as an ordered pair $(\mathtt{V},\mathtt{E})$ as the second writer does, then obviously your very definition of $V$ and $E$ is:

$$V(\mathtt{V},\mathtt{E})=\mathtt{V},\quad E(\mathtt{V},\mathtt{E})=\mathtt{E}.$$

Under these definitions, its clearly the case that $G = (V(G),E(G))$ for all graphs $G$.

Concern 3: wikipedia's definition of graph says edges are "2-element subsets of V" but this definition says the set can be either one or two elements from V. This makes more sense to me because an edge can be a closed loop, and in this case wouldn't it only have 1 vertex? Also the second definition says there are two distinct elements from V that make up an element in ε which makes loops impossible.

Wikipedia's definition coincides with that of the second author, while the first author's definition is strictly more general. In general, terminology in graph theory is not-entirely standardized, so you'll just have to keep your wits about you. Admittedly, this mismatch is a tad annoying, but its not a worthy reason to disregard either book.

Concern 4: why does the second definition use ε instead of E for edges?

The $\varepsilon$ isn't really part of the second definition; its just a dummy variable. If you replace it by $E$, you get the same definition. So this is strictly just stylistic thing; the second author will probably continue to choose $\varepsilon$ to denote the edge set of whatever graph he/she is discussing, but its not an important distinction. Compare with the approach of the first author, where $E$ is actually what's being defined, as opposed to $E'$ or $\varepsilon$ or any other symbol.