Characteristic, admittance and matching polynomials of an antiregular graph

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.

Spectral characterizations of anti-regular graphs

“…, n} is ⌊(n + 1)/2⌋. In [6], the characteristic and matching polynomial of A n are studied and several recurrence relations are obtained for these polynomials, along with some spectral properties of the adjacency matrix of A n .…”

The role of the anti-regular graph in the spectral analysis of threshold graphs

“…The graph A n is a threshold graph with binary string b = 0101 · · · 01 when n is even and b = 00101 · · · 01 when n is odd. It was proved in [18] (see also [1]) that A n has simple eigenvalues and moreover has inertia i(A 2k ) = (k, 0, k) if n = 2k is even and i(A 2k+1 ) = (k, 1, k) if n = 2k + 1 is odd. Thus, λ k+1 (A 2k+1 ) = 0 and A 2k+1 does not contain −1 as an eigenvalue.…”

“…A simple graph G = (V, E) is a threshold graph if there exists a function w : V → [0, ∞) and a real number t ≥ 0 called the threshold such that for every X ⊂ V , X is an independent set if and only if v∈X w(v) ≤ t. Threshold graphs were independently introduced in [5] and [10]; for a comprehensive survey of threshold graphs see [17]. Threshold graphs have applications in resource allocation problems where the weight w(v) is the amount of resources used by vertex v and thus X is an admissible subset of vertices if the total amount of resources required by X is no more than the allowable threshold t. In this paper, we are interested in the eigenvalues of the (0, 1)-adjacency matrix A(G) of a threshold graph G. To the best of the authors' knowledge, the first study on the spectral properties of threshold graphs was focused on a specific threshold graph called the anti-regular graph [18]. In [18], several recurrence relations were obtained for the characteristic polynomial of the unique n-vertex connected anti-regular graph A n , and moreover it was shown that A n has simple eigenvalues with bipartite character.…”

“…Threshold graphs have applications in resource allocation problems where the weight w(v) is the amount of resources used by vertex v and thus X is an admissible subset of vertices if the total amount of resources required by X is no more than the allowable threshold t. In this paper, we are interested in the eigenvalues of the (0, 1)-adjacency matrix A(G) of a threshold graph G. To the best of the authors' knowledge, the first study on the spectral properties of threshold graphs was focused on a specific threshold graph called the anti-regular graph [18]. In [18], several recurrence relations were obtained for the characteristic polynomial of the unique n-vertex connected anti-regular graph A n , and moreover it was shown that A n has simple eigenvalues with bipartite character. Specifically, A n has ⌊ n 2 ⌋ negative and ⌊ n 2 ⌋ positive eigenvalues; when n is odd A n has a zero eigenvalue and when n is even A n has −1 as an eigenvalue.…”

Irregularity of molecular graphs

“…However, Behzad and Chartrand [9] established that perfect graphs do not exist. What does exist are the "quasi-perfect" graphs (sometimes referred to as "antiregular graphs" [43]). These are graphs in which all vertices, except two, have different degrees.…”