IGraph/M, the Mathematica interface to igraph, just had version 0.5 released.
You will find installation/upgrade instructions here.
New functions

IGSplitQ
recognizes split graphs and their degree sequences. 
IGThresholdQ
recognizes threshold graphs and their degree sequences. 
IGPotentiallyConnectedQ
recognizes the degree sequences of connected graphs. 
IGBigraphicalQ
recognizes the degree sequence pairs of bipartite graphs. 
IGEulerianQ
tests if a graph has an Eulerian path;IGEulerianPath
andIGEulerianPathVertices
find it.
Changes to existing functions

IGGraphicalQ
andIGRealizeDegreeSequence
now support some types of nonsimple graphs. 
IGRealizeDegreeSequence
now takes arguments in the orderindegrees
,outdegrees
for consistency with other functions (previously it wasoutdegrees
,indegrees
). 
IGEigenvectorCentrality
sometimes returned incorrect values for isolated vertices in weighted graphs. 
IGColoredSimpleGraph
no longer discards vertex names. 
IGModularity
now supports directed graphs. 
IGModularity
andIGCommunitiesMultilevel
now have a resolution parameter. 
IGAdjacencyMatrixPlot
now allowsNone
to be specified as the colour representing nonexisting edges. 
IGEigenvectorCentrality
assumes the adjacency matrix of undirected graphs to have twice the number of selfloops for each vertex on the diagonal. This makes the results consistent between an undirected graph and its directed equivalent when each edge is replaced by a mutual edge pair.
Fixes

IGLayoutReingoldTilford
no longer flips the layout. 
IGLayoutReingoldTilford
no longer draws overlapping tree branches. 
IGBarabasiAlbertGame
now allows negative values for β. 
IGBetweennessEstimate
sometimes returned incorrect results with finite cutoffs. This is now corrected. 
IGCommunitiesLeiden
: fix incorrect results when selfloops are present. 
IGEigenvectorCentrality
: fix incorrect results for isolated vertices and for vertices with selfloops. 
IGGraphicalQ
would return incorrect results when the second argument was{}
. This is now corrected.
Other changes

IGDegreeSequenceGame
's"ConfigurationModelSimple"
method is now much faster.  More robust error handling.
Notes
 IGraph/M 0.5 has been tested with Mathematica 10.3 and later only. Some effort has been made to allow it to work with Mathematica 10.0.2, but it has not been tested and compatibility is not guaranteed.