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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Let G be a finite simple graph. The line graph L(G) represents the adjacencies between edges of G. We define first the line simplicial complex ∆ L (G) of G containing Gallai and anti-Gallai simplicial complexes ∆ Γ (G) and ∆ Γ ′ (G) (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex ∆ L (G) is connected if and only if G is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes. In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.

A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes ∆ s (G) of multigraphs G, which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a uni-cyclic multigraph U r n,m with n edges including r multiple edges within and outside the cycle of length m. Then, we determine the facet ideal I F (∆ s (U r n,m )) of spanning simplicial complex ∆ s (U r n,m ) and its primary decomposition. The Euler characteristic is a well-known topological and homotopic invariant to classify surfaces. Finally, we device a formula for Euler characteristic of spanning simplicial complex ∆ s (U r n,m ).

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