  We propose a notion of graph convergence that interpolates between the
Benjamini--Schramm convergence of bounded degree graphs and the dense graph
convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that
spectra of graphs, and also some important graph parameters such as numbers of
colorings or matchings, behave well in convergent graph sequences. Special
attention is given to graph sequences of large essential girth, for which
asymptotics of coloring numbers are explicitly calculated. We also treat
numbers of matchings in approximately regular graphs.
  We introduce tentative limit objects that we call graphonings because they
are common generalizations of graphons and graphings. Special forms of these,
called Hausdorff and Euclidean graphonings, involve geometric measure theory.
We construct Euclidean graphonings that provide limits of hypercubes and of
finite projective planes, and, more generally, of a wide class of regular
sequences of large essential girth. For any convergent sequence of large
essential girth, we construct weaker limit objects: an involution invariant
probability measure on the sub-Markov space of consistent measure sequences
(this is unique), or an acyclic reversible sub-Markov kernel on a probability
space (non-unique). We also pose some open problems.
