The Least Possible Eigenvalue of a Super Line Multigraph

Not long ago, Bagga, Beineke, and Varma [1] defined the super line multigraph of a simple graph Γ = (V,E) to be the graph Mr(Γ) whose vertex set is Pr(E), the class of r-element subsets of the edge set, and with an adjacency R ∼ R′ (where R,R′ ∈ Pr(E)) for every edge pair (e, f) with e ∈ R and f ∈ R′ such that e and f are adjacent in Γ. Thus, the number of edges joining R and R′ in Mr(Γ) is the number of such ordered edge pairs (e, f). The simplest example is M|E|(Γ), which has a single vertex but may have many loops. The super line multigraph generalizes the line graph, which equals M1(Γ), but there is the significant difference that when r ≥ 2 the super line multigraph is always connected, as long as Γ has fewer than r components consisting of a single edge. The adjacency matrix A(Mr(Γ)), unlike that of the simplified super line graph (where the multiple edges are reduced to single edges), has good eigenvalue properties; in particular, Bagga and Ferrero [2] proved that its least eigenvalue is not less than −2 ( q−1 r−1 ) , where q := |E| and 1 ≤ r ≤ q. This fact is an elementary extension of a familiar property of the line graph L(Γ); but Bagga, Ellis, and Ferrero went deeper by showing exactly when this minimum really is an eigenvalue. Recall that an eigenvalue of a graph is defined as an eigenvalue of its adjacency matrix. Theorem 1 (Bagga, Ellis, and Ferrero [4, 3]). For 1 ≤ r ≤ q, −2 ( q−1 r−1 ) is an eigenvalue of Mr(Γ) if and only if Γ contains either an even circle or a pair of edge-disjoint odd circles in the same connected component.