Abstract
We introduce first the spanning simplicial complex (SSC) of a multigraph [Formula: see text], which gives a generalization of the SSC associated with a simple graph [Formula: see text]. Combinatorial properties are discussed for the SSC of a family of uni-cyclic multigraphs [Formula: see text] with [Formula: see text] edges including [Formula: see text] multiple edges within and outside the cycle of length [Formula: see text], which are then used to compute the [Formula: see text]-vector and Hilbert series of face ring [Formula: see text] for the SSC[Formula: see text]. Moreover, we find the associated primes of the facet ideal [Formula: see text]. Finally, we device a formula for homology groups of [Formula: see text] and prove that the SSC of a family of uni-cyclic multigraphs is Cohen-Macaulay.
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