  We investigate, using the notion of linear quotients, significative classes
of connected graphs whose monomial edge ideals, not necessarily squarefree,
have linear resolution, in order to compute standard algebraic invariants of
the polynomial ring related to these graphs modulo such ideals. Moreover we are
able to determine the structure of the ideals of vertex covers for the edge
ideals associated to the previous classes of graphs which can have loops on any
vertex. Lastly, it is showed that these ideals are of linear type.
