On the index of cactuses with n vertices

Abstract

Among all connected cactuses with n vertices we find a unique graph whose largest eigenvalue (index, for short) is maximal.

Some graphs determined by their (signless) Laplacian spectra

“…Let W n be the wheel graph on n vertices, i.e., W n = K 1 ∨ C n−1 . A graph is a cactus, or a treelike graph, if any pair of its cycles has at most one common vertex [1], [26]. If all cycles of the cactus G have exactly one common vertex, then G is called a bundle [1].…”

“…A graph is a cactus, or a treelike graph, if any pair of its cycles has at most one common vertex [1], [26]. If all cycles of the cactus G have exactly one common vertex, then G is called a bundle [1]. Let S(n, c, k) be the bundle graph obtained by attaching n − 2c − 2k − 1 pendant edges together with k hanging paths of length two at the vertex v 0 , where v 0 is the unique common vertex of c triangles.…”

“…S(n, c, k) have been investigated in many papers. For instance, S(n, c, 0) is the unique graph with the maximal spectral radius [1] (or the Merrifield-Simmons index [19]), the minimal Hosoya index (or the Wiener index [19], the Randić index [19]) in the set of all connected cacti on n vertices with c cycles, and S(n, 0, β − 1) is the unique tree with the maximum Laplacian Estrada index [8], and the minimum Laplacian-energy-like invariant [15], (or the Wiener index [9], the hyper-Wiener index [28]) in the class of trees with n vertices and the matching number β, where 2 β ⌊ 1 2 n⌋. Moreover, S(n, 1 2 (n − k) − 1, 1) is also an extremal graph [17] with the maximum signless Laplacian spectral radius in the class of connected cacti with n vertices and k pendant vertices.…”

On the spectral radius of cactuses with perfect matchings

“…The maximization problem for cactuses with fixed number of cycles is resolved in [1]. Recall, a cactus is a connected graph in which any two cycles have at most one common vertex.…”

Graphs determined by their (signless) Laplacian spectra

“…For instance, S(n, c) is the graph with the maximal spectral radius, the maximal Merrifield-Simmons index, the minimal Hosoya index, the minimal Wiener index, and the minimal Randić index in the set of all connected cacti on n vertices with c cycles [1,14]. In this paper, by using a new method different from [6,9,15,21,23,24], we show that S(n, c) together with its complement are determined by their Laplacian spectra, and we also prove that S(n, c) together with its complement are determined by their signless Laplacian spectra.…”