Abstract
An antiregular graph is a simple graph with the maximum number of vertices with different degrees. In this paper we study the characteristic polynomial, the admittance (or Laplacian) polynomial and the matching polynomial of a connected antiregular graph. For these polynomials we obtain recurrences and explicit formulas. We also obtain some spectral properties. In particular, we prove an interlacing property for the eigenvalues and we give some bounds for the energy.
Highlights
Any graph on n vertices, with n ≥ 2, has at least two vertices with the same degree
In An the two vertices with the same degree are those with degree n/2
If Gn is the graph with vertices 1, 2, . . . , n where the vertex i is adjacent to the vertex j exactly when i is even and i > j, An = Gn when n is even and An = Gn when n is odd
Summary
Any graph on n vertices, with n ≥ 2 , has at least two vertices with the same degree. The graphs with at most two vertices with the same degree are called antiregular [12, 13], maximally nonregular [20] or quasiperfect [2, 14, 17]. In An two vertices of degree d and e , respectively, are adjacent if and only if d + e ≥ n (see [14]) This simple property gives an iterative procedure to construct the graphs An , starting from the complete graph K1 on one vertex. In this paper we will study the characteristic polynomial, the admittance (or Laplacian) polynomial and the matching polynomial of connected antiregular graphs. First we obtain two recurrences for the characteristic polynomials We obtain their generating series and we show that they can be expressed in terms of Chebyshev polynomials of the first and the second kind. We study the matching polynomial and in this case we give a recurrence, a generating series and some explicit expressions
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