The first book on graph theory was König's Theorie der endlichen und
unendlichen Graphen (Theory of finite and infinite graphs) of 1936.
Thus infinite graphs were part of graph theory from the very beginning.
König's most important result on infinite graphs was the so-called König infinity lemma,
which states that in an infinite, finitely-branching, tree there is an
infinite branch. This lemma encapsulates many arguments -- from the
Bolzano-Weierstrass theorem, to the completeness theorem of logic, to
the proof of various Ramsey theorems --
in graph-theoretic form. König himself used it to prove that the
infinite form of van der Waerden's theorem on arithmetic progressions
implies the finite version, and Erdos and Szekeres (who were students
of König) took up the idea in their pioneering 1935 paper on Ramsey
theory.
As other commentators have mentioned, infinite graphs are also
important as group diagrams in combinatorial group theory and
low-dimensional topology.
As pointed out in comments and at this question, this answer is not entirely correct because real/quaternionic does not always give a $\mathbb{Z}/2$ grading. Nonetheless in practice this answer seems to be "mostly right." I'm going to leave it up unmodified because changing it would confuse the comments and other answers too much. Sorry for the mistake!
If $G$ has a quaternionic representation and $G$ has no complex representations (i.e. reps whose character is complex) then for certain nontrivial central elements the square root function is $\sum_\chi \chi(1)$ which is clearly the maximum possible value and clearly larger than the value on the identity.
The key idea in the proof is the fact that $Z(G)^*$ (the group of characters of the center of $G$) is the universal grading group for the category of $G$ representations.
Let me unpack that a little bit. To every irrep of $G$ you can assign it's "central character", that is an element of $Z(G)^*$. This assignment is multiplicative in the sense if $U$ is a subrep of $V \otimes W$ then the central character of $U$ is the product of the central characters of $V$ and $W$ (this is just because central elements act by scalars). In other words, $\operatorname{Rep}(G)$ is graded by $Z(G)^*$. The claim is that an $H$-grading of $\operatorname{Rep}(G)$ is the same thing as a map $Z(G)^* \to H$.
Why is this true? I'm just going to sketch the proof, in particular I'm just going to talk about the case where $Z(G)$ is trivial, but I'll indicate how the general case works. Since the center is trivial you can find a faithful representation $V$. But then let's look at a really high tensor power of $V$ and ask how it breaks up. We can compute that using character theory. Since the rep is faithful and there's no center, the character of a high tensor power of $V$ is dominated by the value on $1$. Hence any high tensor power of $V$ contains all other irreps. In particular, the $n$-th and $(n+1)$-th powers contain the same irreps and this tells you that the grading group is trivial. In general you want to argue that the contribution of the center dominates the values of inner products (but you need to be a bit careful about non-faithful representations).
The "universal grading group" is used by Gelaki and Nikshych to define the upper central series of an arbitrary fusion category and thereby define nilpotent fusion categories.
Ok, how is this relevant to anything? Well, if you have a group with a quaternionic representation but no complex representations, then the Frobenius-Schur indicator gives a $\pm 1$ grading of $Rep(G)$, namely put the real reps in grade $1$, and the quaternionic ones in grade $-1$. Hence the Frobenius-Schur indicator $s(\chi)$ must be given by the central character $\chi(z)/\chi(1)$ for some central element $z$. Clearly these central elements maximize the square root function.
Best Answer
One group theory topic in which logicians have been active is the automorphism tower problem.
The automorphism tower of a group $G$ is obtained by computing the automorphism group $\newcommand\Aut{\text{Aut}}\Aut(G)$, and then the automorphism group of that group $\Aut(\Aut(G))$ and so on. Each group maps into the next by inner automorphisms, and so one may continue the iteration transfinitely with the direct limit. The main question had been whether this process ever stops, and if so, how long it takes.
Simon Thomas, using an application of Fodor's lemma in set theory, showed that every centerless group $G$ has an automorphism tower that terminates before $(2^{|G|})^+$ steps. He had been writing a book on the topic, which was excellent, although I'm not sure if this is yet fully realized.
Greg Kuperberg wrote a nice summary answer to this classic MathOverflow question, Does $\Aut(\Aut(\cdots\Aut(G))))$ stabilize?.
My 1998 paper Every group has a terminating automorphism tower shows that every group has a terminating tower, and one can find other papers and talks (with slides) about the automorphism tower problem on my blog: jdh.hamkins.org/tag/automorphism-towers.
Perhaps one of the most interesting connection with logic is my joint result with Simon Thomas in our 2000 paper Changing the heights of automorphism towers, where we show that the very same group can have automorphism towers of vastly different heights, depending on the set-theoretic background universe in which the tower is computed.