The Matching Polynomials and Spectral Radii of Uniform Supertrees

Abstract

We study matching polynomials of uniform hypergraph and spectral radii of uniform supertrees. By comparing the matching polynomials of supertrees, we extend Li and Feng's results on grafting operations on graphs to supertrees. Using the methods of grafting operations on supertrees and comparing matching polynomials of supertrees, we determine the first d 2 + 1 largest spectral radii of r-uniform supertrees with size m and diameter d. In addition, the first two smallest spectral radii of supertrees with size m are determined.MSC classification : 15A18, 05C65, 05C31

The largest spectral radius of uniform hypertrees with a given size of matching

“…Then H is said to be obtained by an edge-releasing operation on e at u. We first give some useful results which were proposed in [10]. (a) ϕ(G∪H, x) = ϕ(G, x)ϕ(H, x).…”

“…Recently Su et. al [10] determine the first d 2 + 1 largest spectral radii of r-uniform hypertrees with size m and diameter d In this paper, using the theory of matching polynomial of hypertrees introduced in [10], we determine the largest spectral radius of hypertrees with m edges and given size of matching. The structure of the remaining part of the paper is as follows: In Section 2, we give some basic definitions and results for tensor and spectra of hypergraphs.…”

“…[10]). If T is an uniform hypertree, and T is a proper partial hypergraph of T with V (T ) = V (T ), then T ≺ T .…”

Spectra of weighted uniform hypertrees

“…Obverse that, if we choose the weighting function w such that w(v) = 0 for all v ∈ V (H) and w(e) = 1 for all e ∈ E(H), then µ(H, w, x) is exactly the matching polynomial µ(H, x) of H defined in [25]. For a k-tree T of order n, one may check that µ(T, x) = x n−km(T ) ϕ(T, x).…”

Sharp Bounds on the Spectral Radii of Uniform Hypergraphs concerning Diameter or Clique Number

“…Su et al [8] determined the largest spectral radius of hypertrees with r edges and given size of matching. Xiao et al [9] determined the supertrees with the first two largest spectral radii among all supertrees in the set of m-uniform supertrees with r edges and diameter d. Su et al [10] determined the first ⌊d/2⌋ + 1 largest spectral radius of k-uniform supertrees with size m and diameter d. In addition, the first two smallest spectral radii of supertrees with size m are also determined. For other related results, readers are referred to [11][12][13][14][15][16][17][18][19][20].…”