On the spread of outerplanar graphs

Daniel Gotshall; Megan O’Brien; Michael Tait

doi:10.1515/spma-2022-0164

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On the spread of outerplanar graphs

Daniel Gotshall , Megan O’Brien and Michael Tait

Published/Copyright: March 31, 2022

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From the journal Special Matrices Volume 10 Issue 1

Abstract

The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. We show that for sufficiently large n, the n-vertex outerplanar graph with maximum spread is a vertex joined to a linear forest with Ω(n) edges. We conjecture that the extremal graph is a vertex joined to a path on n−1 vertices.

Keywords: spread; adjacency eigenvalues; outerplanar

MSC 2010: 05C50

1 Introduction

The spread of a square matrix M is defined to be

S(M)≔maxi,j∣λi−λj∣,

where the maximum is taken over all pairs of eigenvalues of M. That is, S(M) is the diameter of the spectrum of M. The spread of general matrices has been studied in several papers (see e.g. [1,2] for two of the early ones). In this article, given a graph G we will study the spread of the adjacency matrix of G, and we will call this quantity the spread of G and denote it by S(G). Since the adjacency matrix of an n-vertex graph is real and symmetric, it has a full set of real eigenvalues which we may order as λ1≥⋯≥λn. In this case, the spread of G is given simply by λ1−λn.

The study of the spread of graphs was introduced in a systematic way by Gregory et al. in [3]. Since then, the spread of graphs has been studied extensively. A problem in this area with an extremal flavor is to maximize or minimize the spread over a fixed family of graphs. This problem has been considered for trees, graphs with few cycles, the family of all n-vertex graphs, bipartite graphs, graphs with a given matching number or girth or size (see the many papers that cite [3] for a long list).

We also note that spreads of other matrices associated with a graph have been considered extensively, but in this article we will focus on the adjacency matrix. This article examines the question of maximizing the spread of an n-vertex outerplanar graph. A graph is outerplanar if it can be drawn in the plane with no crossings and such that all vertices are incident with the unbounded face. Similar to Wagner’s theorem characterizing planar graphs, a graph is outerplanar if and only if it does not contain either K2,3 or K4 as a minor. Maximizing the spread of this family of graphs is motivated by the substantial history on maximizing eigenvalues of planar or outerplanar graphs [4,5,6]; see also [7] and references therein for the history of this problem.

Our main theorem comes close to determining the outerplanar graph of maximum spread. Let Pk denote the path on k vertices and G∨H the combination of G and H. A linear forest is a disjoint union of paths.

Theorem 1.1

Fornsufficiently large, any graph which maximizes spread over the family of outerplanar graphs onnvertices is of the formK1∨F, whereFis a linear forest withΩ(n)edges.

We leave it as an open problem to determine whether or not F should be a path on n−1 vertices, and we conjecture that this is the case.

Conjecture 1.2

Fornsufficiently large, the uniquen-vertex outerplanar graph of maximum spread isK1∨Pn−1.

2 Preliminaries

Let G be an outerplanar graph of maximum spread and let A be its adjacency. We will frequently assume that n is sufficiently large. We will use the characterization that a graph is outerplanar if and only it does not contain K2,3 or K4 as a minor. In particular, G does not contain K2,3 as a subgraph. Given a vertex v∈V(G), the neighborhood of v will be denoted by N(v) and its degree by dv. If f,g:N→R we will use f=O(g) to mean that there exists a constant c such that f(n)≤cg(n) for n sufficiently large. f=Ω(g) means that g=O(f) and f=Θ(g) means that f=O(g) and g=O(f). We will occasionally have sequences of inequalities where we will abuse notation and mix inequality symbols with O(⋅) and Θ(⋅).

Let the eigenvalues of A be represented as λ1≥λ2≥⋯≥λn. For any disconnected graph, adding an edge between the connected components will not decrease λ1 and will also not decrease −λn. Therefore, without loss of generality, we may assume that G is connected. By the Perron-Frobenius theorem we may assume that the eigenvector x corresponding to λ1 has xu>0 for all u.

Furthermore, we will normalize x so that it has maximum entry equal to 1, and let xw=1 where w is a vertex attaining the maximum entry in x. Note that there may be more than one such vertex, in which case we can arbitrarily choose and fix w among all such vertices. The other eigenvector of interest to us corresponds to λn, call it z. We will also normalize z so that its largest entry in absolute value has absolute value 1 and let w′ correspond to a vertex with maximum absolute value in z (so xw′ equals 1 or −1).

We will implement the following known equalities for the largest and smallest eigenvalues:

(2.1)λ1=maxx′≠0(x′)tAx′(x′)tx′=xtAxxtx,

(2.2)λn=minz′≠0(z′)tAz′(z′)tz′=ztAzztz.

An important result from equations (2.1) and (2.2) and the Perron-Frobenius theorem is that for any strict subgraph H of G, we have λ1(A(G))>λ1(A(H)). Finally, we will use the following theorem from [7].

Theorem 2.3

Fornlarge enough, K1∨Pn−1has maximum spectral radius over alln-vertex outerplanar graphs.

3 Vertex of maximum degree

The main goal of this section is to prove that G has a vertex of degree n−1. This is stated in the following theorem.

Theorem 3.1

Fornlarge enough, we havedw=n−1.

As a first step, we will obtain preliminary upper and lower bounds on the largest and smallest eigenvalues of G. First we obtain an upper bound on the spectral radius.

Lemma 3.2

λ1≤n+1.

Proof

We define the graph G1 to be the graph K1∨Pn−1. By Theorem 2.3, we know that any outerplanar graph on sufficiently many vertices cannot have a spectral radius larger than that of G1. Now define G2 as G1 with another edge joining the endpoints of the path, so G2=K1∨Cn−1. Clearly G1 is a subgraph of G2. Putting all this together gives us

λ1(G)≤λ1(G1)<λ1(G2)=n+1,

where the last equality can be calculated using an equitable partition with two parts (the dominating vertex and the cycle).□

Next we bound ∣λn∣.

Lemma 3.3

Fornsufficiently large, n−1−2≤∣λn∣≤n−1+2.

Proof

The upper bound on ∣λn∣ follows immediately from Lemma 3.2 and the well-known fact λ1≥∣λn∣ for any graph. Now to obtain the lower bound, since G is the outerplanar graph on n vertices that maximizes spread, we have

S(G)≥S(K1,n−1)=λ1(K1,n−1)−λn(K1,n−1)=n−1−(−n−1)=2n−1.

So,

2n−1≤λ1(G)−λn(G)≤n+1−λn(G)<n−1+2−λn(G).

Hence, we have

−λn≥n−1−2.□

Essentially, the same proof also gives a lower bound for λ1.

Corollary 3.4

Fornlarge enough we haveλ1≥n−1−2.

We shall use Lemma 3.3 to obtain a lower bound on the degree of each vertex.

Lemma 3.5

Letube an arbitrary vertex inG. Thendu>∣zu∣n−O(n)anddu>xun−O(n).

Proof

We will show the first part explicitly. When λn is the smallest eigenvalue for our graph, we have

∣λn2zu∣=∑y∼u∑v∼yzv≤du+∑y∼u∑v∼yv≠u∣zv∣.

Recall that an outerplanar graph cannot have a K2,3. This implies every vertex in G has at most two neighbors in N(u), meaning the eigenvector entry for each vertex contained in the neighborhood of N(u) can be counted at most twice. Hence, ∑y∼u∑v∼yv≠u∣zv∣≤2∑v≠u∣zv∣. Note

∣λn∣∣zv∣≤∑v′∼v∣zv′∣≤∑v′∼v1=dv.

So we have

∑y∼u∑v∼yv≠u∣zv∣≤2∑v≠u∣zv∣≤2∣λn∣∑v≠udv≤4e(G)∣λn∣≤4(2n−3)∣λn∣,

as e(G)≤2n−3 by outerplanarity. Combining and using Lemma 3.3, we have

(n−1−2)2∣zu∣≤∣λn2zu∣≤du+4(2n−3)∣λn∣≤du+8nn−1−2,

for n sufficiently large. Isolating du gives the result. A similar proof can be written to justify the lower bound with respect to x and we omit these details.□

Lemma 3.6

We havedw>n−O(n)anddw′>n−O(n). For every other vertexuwe obtain∣zu∣,xu=O1n, for n sufficiently large.

Proof

The bound on dw and dw′ follows immediately from the previous lemma and the normalization that ∣zw′∣=xw=1. Now consider any other vertex u. We know that G contains no K2,3 and hence can have at most two common neighbors with w. Thus, du=O(n). By Lemma 3.5, we have that there are constants c1 and c2 such that

c1n>nxu−c2n

and

c1n>n∣zu∣−c2n,

for sufficiently large n. This implies the result.□

If w and w′ were distinct vertices, then for sufficiently large n they would share many neighbors, contradicting outerplanarity. Hence, we immediately have the following important fact.

Corollary 3.7

Fornsufficiently large we havew=w′.

Hence from now on, we will denote the vertex in Corollary 3.7 by w, and furthermore for the remainder of the article we will also assume that zw=1 without loss of generality. Before deriving our next result quantifying the other entries of z, we first need to define an important vertex set.

Definition 3.8

Recall that w is the fixed vertex of maximum degree in G. Let B=V(G)\(N(w)∪{w}).

Now we consider the zu eigenvector entries of vertices in B. At the moment we know that each eigenvector entry for a vertex in B has order at most 1n. The next lemma shows that in fact the sum of all of the eigenvector entries of B has this order.

Lemma 3.9

Fornlarge enough, we have that∑u∈B∣zu∣and∑u∈Bxuare eachO1n.

Proof

Let u∈B. Since u is not adjacent to w, all of the neighbors of u have eigenvector entry of order at most 1/n and the size of B is also of order at most n by Lemma 3.6. Hence, there is a constant C such that

∣λn∣∣zu∣≤∑v∼u∣zv∣≤Cdun,

and ∣B∣≤Cn. Now

∑u∈B∣zu∣≤1∣λn∣∑u∈BCdun≤C∣λn∣n(e(B,V(G)⧹B)+2e(B)).

Each vertex in B is connected to at most two vertices in N(u), so e(B,V(G)⧹B)≤2∣B∣≤2Cn. The graph induced on B is outerplanar, so e(B)≤2∣B∣−3<2Cn. Finally, using Lemma 3.3, we obtain the required result. A slightly modified version of this argument proves the bound on ∑u∈Bxu.□

We will use Lemma 3.9 to show that B is empty, and this will complete the proof of Theorem 3.1. First we define the following alteration of G. Let t be an arbitrary vertex in B (if it exists).

Definition 3.10

Let G∗ be the graph defined such that its adjacency matrix A∗ satisfies

Aij∗=1{i,j}={w,t}0{i,j}={u,t}whereu≠wAijotherwise.

That is, to obtain G∗ from G we add an edge from t to w and remove all other edges incident with t. In particular, the only neighbor of the t vertex in G∗ is w.

Lemma 3.11

For large enoughn, Bis empty.

Proof

Assume for contradiction that B is nonempty. Then there is a vertex t such that t≁w. Define G∗ as in Definition 3.10. Furthermore, we define the vector z∗ which slightly modifies z as follows.

zu∗=zuifu≠t−∣zu∣ifu=t.

That is, if zt<0, then z∗ is the same vector as z and otherwise we flip the sign of zt. Note that (z∗)Tz∗=zTz. Now

S(G∗)−S(G)≥xTA∗xxTx−(z∗)TA∗z∗zTz−xTAxxTx−zTAzzTz=2xtxTx1−∑v∼txv+2ztzTzsgn(zt)+∑v∼tzv≥2xtxTx1−∑v∼txv+2∣zt∣zTz1−∑v∼tzv,

where sgn(zt) equals 1 if zt>0 and −1 otherwise.

∑v∼tzv≤∑v∼t∣zv∣≤∑v∼tv∉B∣zv∣+∑v∈B∣zv∣.

There are at most two terms in the first sum, and so by Lemmas 3.6 and 3.9, we have

∑v∼tzv=O1n.

Similarly, we have

∑v∼txv=O1n.

This implies 1−∑v∼txv>0 and 1−∑v∼tzv>0 for n large enough, which implies that

2xtxTx1−∑v∼txv+2∣zt∣zTz1−∑v∼tzv>0

for n large enough. Hence S(G∗)>S(G), contradicting the assumption that G is spread-extremal.□

We have finally achieved our goal for this section. Theorem 3.1 follows immediately from the definition of B and the fact that it is empty, as implied by Lemma 3.11.

4 Determining graph structure

By Theorem 3.1, the vertex w has degree n−1, or equivalently, K1,n−1 is a subgraph of G. Since G is K2,3-free, the graph induced by the neighborhood of w has maximum degree at most 2. Furthermore, this subgraph cannot contain a cycle, otherwise G would contain a wheel graph and this is a K4-minor. Any graph of maximum degree at most 2 that does not contain a cycle is a disjoint union of paths. Therefore, we know that G is given by a K1∨F, where F is a disjoint union of paths. Our next task is to study F. To this end, we denote the number of edges in F by m. Our main theorem, Theorem 1.1 is proved if we can show that m=Ω(n). Before doing this, we need a more accurate estimate for the eigenvector entries.

Lemma 4.1

For anyu≠w, we havezu=1λn+du−1λn2+Θ1n3/2.

Proof

As we are only considering outerplanar graphs, we have du∈{1,2,3} for all u≠w. Hence,

λnzu=∑y∼uzy=zw+∑y∼uy≠wzy=1−Θ1n

by Lemma 3.6 and our normalization.

Note that as λnzu=1+zu1+zu2>0 and λn<0, we must have zu<0. Next we repeat the argument to improve our estimate,

λnzu=zw+∑y∼uy≠wzy=1+(du−1)1λn+Θ1n.

Now we apply our bounds on λn in Lemma 3.3 to obtain

zu=1λn+du−1λn2+Θ1n3/2.□

We can use equivalent reasoning to obtain a very similar approximation for the xu entries.

Lemma 4.2

For anyu≠w, we havexu=1λ1+du−1λ12+Θ1n3/2.

Proof

By the same logic as in the proof of Lemma 4.1, we have

λ1xu=∑w∼uxw=xw+∑y∼uy≠wxy=1+Θ1nby Corollary 3.6.

So xu=1λ1+Θ1n. Next we repeat the argument to improve our estimate,

λ1xu=xw+∑y∼uy≠wxy=1+(du−1)1λ1+Θ1n.

So xu=1λ1+du−1λ12+Θ1n3/2 according to our bounds on λ1 in Lemma 3.2.□

Using Lemma 4.1, we can now obtain tighter estimates on λn.

Lemma 4.3

We have thatλn=−n−1+mn−1+Θmn3/2.

Proof

We first define the vector

y2=1−1n−1−1n−1⋮−1n−1,

where y2 has n entries and w corresponds to the 1 entry. As λn is the minimum Rayleigh quotient we have

λn≤y2TAy2y2Ty2=2∑i∼j(y2)i(y2)j2=(n−1)−1n−1+m1n−1=−n−1+mn−1.

In order to show the lower bound on λn, we need to realize that λn is the minimum possible Rayleigh quotient. So

λn=zTAzzTz=2∑i∼jzizjzTz=2∑w∼kzwzkzTz+2∑i∼j,i,j≠wzizjzTz.

Note the first term is the Rayleigh quotient for the star subgraph centered at the vertex w of maximum degree. Hence, it is bounded from below by −n−1. More specifically, we have

2∑w∼kzwzkzTz=zTA(K1,n−1)zzTz≥λn(A(K1,n−1))=−n−1.

Applying Lemma 4.1 we have

λn≥−n−1+2∑i∼ji,j≠w1λn+di−1λn2+Θ1n3/21λn+dj−1λn2+Θ1n3/21+∑ℓ≠w1λn+dℓ−1λn2+Θ1n3/22.

For ℓ≠w we have 0≤dℓ−1≤2. Note that for n large enough we have that 1λn+du−1λn2+Θ1n3/2<0, and so we have a lower bound if we replace all dℓ−1 terms in the numerator by 2 and in the denominator by 0, and so we have

λn≥−n−1+2∑i∼ji,j≠w1λn+2λn2+Θ1n3/21λn+2λn2+Θ1n3/21+∑ℓ≠w1λn+Θ1n3/22=−n−1+2m1λn2+Θ1n3/21+(n−1)1λn2+Θ1n3/2.

Since −n−1−2≤λn≤−n−1+2, we have that 1λn2=1n−1+Θ1n3/2. Hence, we have

λn≥−n−1+2m1n−1+Θ1n3/21+(n−1)1n−1+Θ1n3/2=−n−1+2mn−1+Θmn3/22+Θ1n1/2=−n−1+mn−1+Θmn3/21+Θ1n,

which completes the proof.□

We claim a very similar result for λ1.

Lemma 4.4

λ1=n−1+mn−1+Θmn3/2.

Proof

First we prove the lower bound. Define the vector

y3=11n−11n−1⋮1n−1,

where w corresponds to the entry 1. As λ1 is the maximum possible Rayleigh quotient we have

λ1≥y3TAy3y3Ty3=2∑i∼j(y3)i(y3)j2=(n−1)1n−1+m1n−1=n−1+mn−1.

For the upper bound, we use the fact that λ1 is the maximum possible Rayleigh quotient similar to the previous lemma. So

λ1=xTAxxTx=2∑i∼jxixjxTx=2∑w∼kxwxkxTx+2∑i∼ji,j≠wxixjxTx.

As before, the first term is

xTA(K1,n−1)xxTx≤λ1(A(K1,n−1))=n−1.

Applying Lemma 4.2 and using 0≤du−1≤2 for all u≠w we have

λ1≤n−1+2∑i∼ji,j≠w1λ1+di−1λ12+Θ1n3/21λ1+dj−1λ12+Θ1n3/21+∑ℓ≠w1λ1+dℓ−1λ12+Θ1n3/22≤n−1+2∑i∼ji,j≠w1λ1+2λ12+Θ1n3/21λ1+2λ12+Θ1n3/21+∑ℓ≠w1λ1+Θ1n3/22.

Using n−1−2≤λ1≤n−1+2, we have

λ1≤n−1+2m1n−1+Θ1n3/21+(n−1)1n−1+Θ1n3/2=n−1+2mn−1+Θmn3/22+Θ1n1/2=n−1+mn−1+Θmn3/21+Θ1n,

which completes the proof.□

We are now in a position to prove our main theorem.

Proof of Theorem 1.1

As before, let G1 be the graph K1∨Pn−1 and let A1 be its adjacency matrix with the dominating vertex corresponding to the first row and column. Since G is spread-extremal we must have S(G)≥S(G1). We lower bound S(G1) using the vectors

v1=−1+n11⋮1andv2=−1−n11⋮1.

Using these vectors and the Rayleigh principle, we have that

S(G1)≥v1TA1v1v1Tv1−v2TA1v2v2Tv2=2((n−1)(n−1)+(n−2)(1))2n−2n−2((n−1)(−n−1)+(n−2)(1))2n+2n=2((n−1)(n−1)+(n−1)(1))2n−2n−22n−2n−2((n−1)(−n−1)+(n−1)(1))2n+2n+22n+2n=n−1n−1+n−1n+1−2nn2−n≥2n−1n,

where the last inequality holds for n>6. Combining this with Lemmas 4.3 and 4.4, we have

2n−1n≤λ1(G)−λn(G)=n−1+mn−1+Θmn3/2−−n−1+mn−1+Θmn3/2.

Therefore, there exists a constant C such that for n large enough we have

2n−1n≤2n−1+C⋅mn3/2.

Since n−n−1=Ω(n−1/2), rearranging gives the final result.□

Acknowledgements

The authors would like to thank the Community of Mathematicians and Statisticians Exploring Research (Co-MaStER) program at Villanova, through which this research was started.

Funding information: Michael Tait was partially supported by National Science Foundation grant DMS-2011553.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2021-11-23

Revised: 2022-03-03

Accepted: 2022-03-09

Published Online: 2022-03-31

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https://doi.org/10.1515/spma-2022-0164

Keywords for this article

spread; adjacency eigenvalues; outerplanar

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