Laplacian spectral characterization and adjacency spectral ratio of four-rose graphs

Let [Formula: see text] be a simple connected graph having [Formula: see text] vertices and [Formula: see text] edges. The Adjacency spectral ratio [Formula: see text] is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the largest and second smallest adjacency eigenvalues for the graph [Formula: see text]. In this paper, we consider the class of 4-rose graphs, consisting of four cycles intersecting in exactly one common vertex. Explicit expressions for the coefficients of the Laplacian characteristic polynomial [Formula: see text] of 4-rose graphs are obtained via spanning forest techniques. Using these expressions, we prove that every 4-rose graph is uniquely determined by its Laplacian spectrum. In addition, we derive new lower bounds for the adjacency spectral ratio in terms of basic degree based invariants, including the first Zagreb index [Formula: see text]. As a further consequence, we establish an upper bound on the number of edges of triangle free 4-rose graphs.