Ansys Defibrillation Threshold

J. Lazzarin, O.F. M´Arquez And F. Tura

Departamento de Matem´atica, UFSM, Santa Maria, RS, 97105-900, Brazil

Abstract

A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the characteristic polyno- mial of a threshold graph from its binary sequence. Applications include obtaining a formula for the determinant of adjacency matrix of a threshold graph and showing that no two nonisomorphic threshold graphs are cospectral.

keywords: threshold graph, characteristic polynomial, cospectral graphs. AMS subject classification: 15A18, 05C50, 05C85.

Introduction

Let G = (V, E) be an undirected graph with vertex set V and edge set E, without loops or multiple edges. The adjacency matrix of G, denoted by A = [aij], is a matrix whose rows and columns are indexed by the vertices of G, and is defined to have entries aij = 1 if and only if vi is adjacent to vj, and aij = 0 otherwise. The characteristic polynomial of G, denoted by PG(x), can be expressed as PG(x) = det(A −xI). Their roots are called eigenvalues of G, or simply the spectrum of G.

This paper is concerned with threshold graphs, introduced by Chv´atal and Hammer and Henderson and Zalcstein in 1977. They are an important class of graphs because of their numerous applications and have several alternative characterizations. For example, a threshold graph is a {2K2, C4, P4}-free graph, that is, a graph which does not contain 2K2, C4, and P4 as an induced subgraph. In particular, a threshold graph has a recursive definition based on a binary sequence that will be relevant to this paper, and we will describe in the next Section.

There is a considerable body of knowledge on the spectral properties of threshold graphs [1, 2, 5, 10–12, 15, 17, 18]. However, in the literature, with exception of some par- ticular cases does not seem to provide many results about formulas for the characteristic polynomial of graphs. In this paper we attempt to fill this gap, presenting in Section 3, an explicit formula to compute the characteristic polynomial of threshold graph from its binary sequence.

Two nonisomorphic graphs with the same spectrum are called cospectral. In recent years, there has been a growing interest to find families of cospectral graphs.

There

are many constructions in the literature [9, 16]. This notion is originally defined for the adjacency matrix of the graph G, but a natural extension of the problem is to find families of graphs that are cospectral with relation to other matrices. For instance, a family of 2n−4 threshold graphs on n ≥4 vertices, which are Q-cospectral was exhibited in , where Q is the signless Laplacian matrix.

The result in motivates us to investigate further the cospectrality related to adja- cency matrix. It seemed to us that cospectrality was present in the threshold graph class. Indeed, in this paper we prove that no two threshold graphs are cospectral. The proof is based on an explicit formula to compute the characteristic polynomial of threshold graphs. We also use this formula to obtain the determinant of its adjacency matrix.

The paper is organized as follows.

In Section 2 We Show The Representation Of A

threshold graph by binary sequence, and some known results. In Section 3 we present an explicity formula to compute the characteristic polynomial of threshold graphs from its binary sequence. We finalize the paper showing in Section 4 that no two threshold graphs are cospectral.

Preliminaries

Recall that a vertex is isolated if it has no neighbors, and is dominating if it is adjacent to all others vertices. A threshold graph is obtained through an iterative process which starts with an isolated vertex, and where, at each step, either a new isolated vertex is added, or a dominating vertex is added.

An alternative way to describe the above inductive construction is to associate a threshold graph G of order n with a binary sequence of length n, where a vertex vi corresponds to digit 0, if an isolated vertex vi is added and vi corresponds to digit 1, if vi was added as a dominating vertex. The choice of digit associated to v1 is arbitrary and we use it as 0. Hence, for a connected threshold graph G with n ≥2 vertices the last vertex of G is associated with a digit of type 1.

Let G = (0a11a2 . . . 0an−11an) denote a connected threshold graph G where each ai is a positive integer. Let Vi denote the set of vertices corresponding to ai (i = 1, 2, . . . , n). Hence, |Vi| = ai. It is easy to check that the partition of set vertex of G into non-empty subsets V1, V2, . . . , Vn determines a divisor H of G . As illustration the Figure 1 shows a threshold graph G = (02130312) and its partitioned representation. The n×n adjacency

A4

. . .

A4

. . .

A4

. . .

A4 −1

. . .

An

... ... ...

... ... ...

...

0

0

0

. . .

A3

. . .

(1)

The following theorem (see, for example ) will play an important role in the sequel. Theorem 1 For a connected threshold graph G = (0a11a2 . . . 0an−11an), let λ be an eigen- value of G, distinct from 0 or −1. If H is a divisor of G then λ appears in the spectrum of H.

We finish this section, presenting an interesting result about the multiplicity of eigen- values 0 and −1 of a threshold graph G. Let m0(G) and m−1(G) denote the multiplicity of the eigenvalue 0 and −1, respectively. According , they can be obtained directly

From Its Binary Sequence, That Is

Theorem 2 For a connected threshold graph G = (0a11a2 . . . 0an−11an) where each ai is a

2

i=1(a2i−1 −1).

I=1(A2I −1)

if a1 = 1.

An Explicity Formula For Pg(X)

In this section we present an explicit formula for the characteristic polynomial of threshold graph from its binary sequence. As first application, we obtain an expression to the determinant of adjacency matrix of threshold graph similar to result presented in .

Theorem 3 Let G = (0a11a2 . . . 0an−11an) be a connected threshold graph and let H be divisor of G. For the n × n adjacency matrix D of H and an indeterminate x, the D −xI matrix can be reduced to tridiagonal matrix of form

10

Figure 1: A threshold graph and its partitioned representation

(2)

Proof: Let D be the matrix of H given by (1). We will prove the case n even, the case n odd follows a similar way. For the matrix D −xI, we first perform on the even rows with following operations Ri ←Ri −Ri−2, for each i = n, n −2, n −4, . . . , 4, giving the following matrix:

A4

. . .

A4

. . .

A4

. . .

−X −1

... ... ...

... ... ...

... ...

0

0

0

0

0

. . .

0

0

0

. . .

(3)

The next step is to operate on the odd rows of matrix (3). Then perform the operations Ri ←Ri −Ri−1, for each i = n −1, n −3, . . . , 3, followed by operation R2 ←R2 −R1,

A4

. . .

0

0

. . .

0

. . .

−X −1

... ... ...

... ... ...

... ...

0

0

0

0

0

. . .

0

0

0

0

. . .

(4)

For cancel the elements in the first row of matrix (4), with the exception of the last two elements, we perform the following operations R1 ←R1 + Ri, for each i = 3, 5, . . . , n −1,

0

0

0

0

. . .

0

0

. . .

0

. . .

−X −1

... ... ...

... ... ...

... ...

0

0

0

0

0

. . .

0

0

0

0

. . .

(5)

The tridiagonal matrix (2) is finally obtained after a permutation on the rows of matrix (5), followed by a multiplication by −1 on the last row.

Let Qn(X) = Qn

i=1(x −λi) denote the polynomial whose roots are the eigenvalues of a divisor H of G = (0a11a2 . . . 0an−11an). Follows immediately from Theorem 3 that the

(6)

since that the determinant of tridiagonal matrix (2) may be easily computed by performing a Laplace expansion. In order to obtain an explicit formula to PG(x) for G = (0a11a2 . . . 0an−11an) we will explore the recursive formula (6).

Let [n] = {1, 2, . . . n}, and let In,l the set of increasing sequences in [n] of length l such that if (t1, t2, . . . , tl) ∈In,l then ti ≡n + i −l (mod 2). In other words, elements in In,l are increasing sequences alternating even and odds numbers such that the last term has the

Same Than Parity N. For Instance

I7,4 = {(2, 3, 4, 5), (2, 3, 4, 7), (2, 3, 6, 7), (2, 5, 6, 7), (4, 5, 6, 7)},

While

I6,4 = {(1, 2, 3, 4), (1, 2, 3, 6), (1, 2, 5, 6), (1, 4, 5, 6), (3, 4, 5, 6)}. In general, for any l the sequences in I7,l must finish in an odd number, while all the sequences in I6,l must finish in an even number. Given a sequence t = (t1, t2, . . . , tl) we denote at = at1at2 · · ·atl. Based in this notation we have Definition 1 Let a1, a2, . . . , an be fixed sequence of positive integers. We define the fol-

Anγn−1(L −1)

if l ∈{n −1, n}. Proof: For 1 ≤l ≤n −2, if t = (t1, t2, . . . , tl) ∈In,l and tl̸ = n then the terms in such sequence are less than n −2, so then in this case t ∈In−2,l. Otherwise, tl = n and 1 ≤ti ≤n −1 satisfies ˆt = (t1, t2, . . . , tl−1) ∈In−1,l−1.

At

= anγn−1(l −1) + γn−2(l). In the case l ∈{n−1, n} it is clear that γn(l) = anγn−1(l−1).

□

We write n = 2m + r0, with r0 ∈{0, 1} and define r1 ∈{0, 1} such that r1 ≡

K=0

(−1)m−kxkykγn(n −2k −r1).

Theorem 4 For N ≥4

pn(x, y) = (−1)n+1an pn−1(x, y) + xy pn−2(x, y).

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