Line-graph inertia of roses and generalized theta graphs

For a graph G, the adjacency inertia of the line graph L(G) is determined by the number of eigenvalues of the signless Laplacian Q(G) above, equal to, and below 2. We compute the inertia of Q(G) - 2I, and hence the inertia of the adjacency matrix of L(G), exactly, including every singular case, for rose graphs and generalized theta graphs.

Both computations follow from a common reduction. Deleting the common vertices leaves disjoint paths. Range-kernel elimination then leaves their singular kernel directions and a residual scalar for a rose graph, or a 2 x 2 matrix for a generalized theta graph. Terminal exchange determines the eigenvectors of the latter. The resulting formulas depend only on the path lengths modulo 4.

We obtain closed expressions for the full inertia, the signature, and the multiplicity of the signless Laplacian eigenvalue 2. One theta mode is the rose scalar shifted by the contribution of the second terminal, while the other has no rose analogue.

Further consequences include the bound m_Q(G,2) <= c(G) for generalized theta graphs with at least three paths, and an exact comparison with the conjectured bound 2s(L(G)) <= c(G) + 1. Its slack grows linearly with the cyclomatic number on both graph classes, with equality only for cycles whose length is congruent to 1 modulo 4. The general conjecture is not proved.

We also give a partial extension to bridgeless cacti. Exact finite computations check every formula branch, and the accompanying archive provides the source files, verification scripts, frozen outputs, and build information.