# Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Authors: Sezer Sorgun
Journal: International Journal of Combinatorics
 (2013)

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## Abstract

LetG be weighted graphs, as the graphs where the edge weights
are positive definite matrices. The Laplacian eigenvalues of a graph are the
eigenvalues of Laplacian matrix of a graph G. We obtain two upper bounds
for the largest Laplacian eigenvalue of weighted graphs and we compare these
bounds with previously known bounds.


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## Body

## 1. Introduction

LetG=(V,E) be simple graphs, as graphs which have no loops or parallel edges such that V is a finite set of vertices and E is a set of edges.A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weight matrices are assumed to be of same order and to be positive matrix. In this paper, by “weighted graph” we mean “a weighted graph with each of its edges bearing a positive definite matrix as weight,” unless otherwise stated.The notations to be used in paper are given in the following.LetG be a weighted graph on n vertices. Denote by wi,j the positive definite weight matrix of order p of the edge ij, and assume that wij=wji. We write i~j if vertices i and j are adjacent. Let wi=∑j:j~iwij. be the weight matrix of the vertex i.The Laplacian matrix of a graphG is defined as L(G)=(lij), where
(1)li,j={wi;ifi=j,-wij;ifi~j,0;otherwise.The zero denotes thep×p zero matrix. Hence L(G) is square matrix of order np. Let λ1 denote the largest eigenvalue of L(G). In this paper we also use to avoid the confusion that ρ1(wij) is the spectral radius of wij matrix. If V is the disjoint union of two nonempty sets V1 and V2 such that every vertex i in V1 has the same ρ1(wi) and every vertex j in V2 has the same ρ1(wj), then G is called a weight-semiregular graph. If ρ1(wi)=ρ1(wj) in weight semiregular graph, then G is called a weighted-regular graph.Upper and lower bounds for the largest Laplacian eigenvalue for unweighted graphs have been investigated to a great extent in the literature. Also there are some studies about the bounds for the largest Laplacian eigenvalue of weighted graphs [1–3]. The main result of this paper, contained in Section 2, gives two upper bounds on the largest Laplacian for weighted graphs, where the edge weights are positive definite matrices. These upper bounds are attained by the same methods in [1–3]. We also compare the upper bounds with the known upper bounds in [1–3]. We also characterize graphs which achieve the upper bound. The results clearly generalize some known results for weighted and unweighted graphs.


## 2. The Known Upper Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

In this section, we present the upper bounds for the largest Laplacian eigenvalue of weighted graphs and very useful lemmas to prove theorems.Theorem 1 (Horn and Johnson [4]).
LetA∈Mn be Hermitian, and let the eigenvalues of A be ordered such that λn≤λn-1≤⋯≤λ1. Then,
(2)λnxTx≤xTAx≤λ1xTx(3)λmax=λ1=maxx≠0xTAxxTx=maxxTx=1xTAxλmin=λn=minx≠0xTAxxTx=minxTx=1xTAx
for all x∈ℂn.Lemma 2 (Horn and Johnson [4]).
LetB be a Hermitian n×n matrix with eigenvalues λ1≥λ2≥⋯≥λn; then for any x-∈Rn(x-≠0-), y-∈Rn(y-≠0-),
(4)|x-TBy-|≤λ1x-Tx-y-Ty-.
Equality holds if and only if x- is an eigenvector of B corresponding to λ1 and y-=αx- for some α∈R.Lemma 3 (see [1]).
LetG be a (ρ1(wi),ρ1(wj))-semiregular bipartite graph of order n such that the first l vertices of the same largest eigenvalue ρ1(wi) and the remaining m vertices of the same largest eigenvalue ρ1(wj). Also let x- be a common eigenvector of wij corresponding to the largest eigenvalue ρ1(wij) for all i, j, where wi=∑k∈Niwik for all i. Then ρ1(wi)+ρ1(wj) is the largest eigenvalue of L(G) and the corresponding eigenvector is
(5)(ρ1(wi)x-T,…,ρ1(wi)x-T︸l,-ρ1(wj)x-T,…,-ρ1(wj)x-T︸m).Theorem 4 (see [1]).
LetG be a simple connected weighted graph. Then
(6)λ1≤maxi~j{ρ1(∑‍k:k~iwik)+∑‍k:k~jρ1(wjk)},
where wij is the positive  definite weight matrix of order p of the edge ij. Moreover equality holds in (6) if and only if(i)
Gis a weight-semiregular bipartite graph,(ii)
wij have a common eigenvector corresponding to the largest eigenvalue ρ1(wij) for all i, j.Theorem 5 (see [2]).
LetG be a simple connected weighted graph. Then
(7)λ1≤maxi~j{∑‍k:k~iρ1(wik)(∑‍r:r~iρ1(wir)+∑‍s:s~kρ1(wks))+∑‍k:k~jρ1(wjk)(∑‍r:r~jρ1(wjr)+∑‍s:s~kρ1(wks))},
where wij is the positive definite weight matrix of order p of the edge ij. Moreover equality holds in (7) if and only if(i)
G is a bipartite semiregular graph;(ii)
wij have a common eigenvector corresponding to the largest eigenvalue ρ1(wij) for all i, j.Corollary 6 (see [2]).
LetG be a simple connected weighted graph where each edge weight wij is a positive number. Then
(8)λ1≤maxi{2wi(wi+w-i)},
where w-i=(∑k:k~iwikwk)/wi and wi is the weight of vertex i. Moreover equality holds if and only if G is a bipartite regular graph.Corollary 7 (see [2]).
LetG be a simple connected weighted graph where each edge weight wij is a positive number. Then
(9)λ1≤maxi~j{wi(wi+w-i)+wj(wj+w-j)},
where w-i=(∑k:k~iwikwk)/wi and wi is the weight of vertex i. Moreover equality holds if and only if G is a bipartite semiregular graph.Theorem 8 (see [2]).
LetG be a simple connected weighted graph. Then
(10)λ1≤maxi~j{ρ1(wi)+ρ1(wj)+(ρ1(wi)-ρ1(wj))2+4γ-iγ-j2},
where γ-i=(∑k:k~iρ1(wik)ρ1(wk))/ρ1(wi) and wij is the positive definite weight matrix of order p of the edge ij. Moreover equality holds in (10) if and only if(i)
G is a weighted-regular graph or G is a weight-semiregular bipartite graph;(ii)
wij have a common eigenvector corresponding to the largest eigenvalue ρ1(wij) for all i, j.Corollary 9 (see [2]).
LetG be a simple connected weighted graph where each edge weight wij is a positive number. Then
(11)λ1≤maxi{wi+w-i},
where w-i=(∑k:k~iwikwk)/wi and wi is the weight of vertex i. Moreover equality holds if and only if G is a bipartite semiregular graph or G is a bipartite regular graph.Theorem 10 (see [3]).
LetG be a simple connected weighted graph. Then
(12)λ1≤maxi{ρ12(wi)+∑‍k:k~iρ12(wik)+∑‍k:k~iρ1(wiwik+wikwk)+∑1≤i,t≤n∑s∈Ni∩Ntρ1(wiswst)},
where wik is the positive definite weight matrix of order p of the edge ik and Ni∩Nk is the set of common neighbours of i and k. Moreover equality holds in (12) if and only if(i)
G is a weight-semiregular bipartite graph;(ii)
wik have a common eigenvector corresponding to the largest eigenvalue ρ1(wik) for all i, k.Corollary 11 (see [3]).
LetG be a simple connected weighted graph where each edge weight wij is a positive number. Then
(13)λ1≤maxi{wi2+∑‍k:k~iwik2+∑‍k:k~i(wiwik+wkwik)+∑‍1≤i,t≤n∑‍s∈Ni∩Ntwiswst}.
Moreover equality holds if and only if G is a bipartite semiregular graph.


## 3. Two Upper Bounds on the Largest Laplacian Eigenvalue of Weighted Graphs

In this section we present two upper bounds for the largest eigenvalue of weighted graphs and compare the bounds with some examples.Theorem 12.
LetG be a simple connected weighted graph. Then(14)λ1≤maxi~j{ρ1(wi)+ρ1(wj)+(ρ1(wi)-ρ1(wj))2+4(∑‍k:k~iρ1(wik))(∑‍k:k~jρ1(wjk))2},where wij is the positive definite weight matrix of order p of the edge ij. Moreover equality holds in (14) if and only if(i)
G is a weighted-regular graph or G is a weight-semiregular bipartite graph;(ii)
wij have a common eigenvector corresponding to the largest eigenvalue ρ1(wij) for all i, j.Proof.
LetX-=(x-1,x-2,…,x-n)T be an eigenvector corresponding to the largest eigenvalue λ1 of L(G). We assume that x-i is the vector component of X- such that
(15)x-iTx-i=maxk∈V{x-kTx-k}.
Since X- is nonzero, so is x-i. Let
(16)x-jTx-j=maxk:k~i{x-kTx-k}
be. The (i,j)th element of L(G) is
(17){wi;ifi=j-wi,j;ifi~j0;otherwise.
We have(18)L(G)X-=λ1X-.
From theith equation of (18), we have
(19)(λ1Ip,p-wi)x-i=-∑‍k:k~iwikx-k,
that is,
(20)x-iT(λ1Ip,p-wi)x-i=-∑‍k:k~ix-iTwikx-k(21)≤∑‍k:k~i|x-iTwikx-k|(22)≤∑‍k:k~iρ1(wik)x-iTx-ix-kTx-kby(4)(23)≤∑‍k:k~iρ1(wik)x-iTx-ix-jTx-jby(16).
From (23) we have
(24)(λ1-ρ1(wi))x-iTx-i≤x-iT(λ1Ip,p-wi)x-iby(2)≤∑‍k:k~iρ1(wik)x-iTx-ix-jTx-j,
that is,
(25)(λ1-ρ1(wi))x-iTx-i≤∑‍k:k~iρ1(wik)x-iTx-ix-jTx-j.
From thejth equation of (18), we get
(26)(λ1Ip,p-wj)x-j=-∑‍k:k~jwjkx-k,
that is,
(27)x-jT(λ1Ip,p-wi)x-j=-∑‍k:k~jx-jTwjkx-k(28)≤∑‍k:k~j|x-jTwjkx-k|(29)≤∑‍k:k~jρ1(wjk)x-jTx-jx-kTx-kby(4)(30)≤∑‍k:k~jρ1(wjk)x-iTx-ix-jTx-jby(15).
Similarly, from (30) we get
(31)(λ1-ρ1(wj))x-jTx-j≤x-jT(λ1Ip,p-wj)x-jby(2)≤∑‍k:k~jρ1(wjk)x-iTx-ix-jTx-j,
that is,
(32)(λ1-ρ1(wj))x-jTx-j≤∑‍k:k~jρ1(wjk)x-iTx-ix-kTx-k.
So, from (25) and (32) we have
(33)(λ1-ρ1(wj))(λ1-ρ1(wi))x-jTx-jx-iTx-i≤(∑‍k:k~iρ1(wik))·(∑‍k:k~jρ1(wjk))x-jTx-jx-iTx-i.
Hence we get(34)(λ1-ρ1(wj))(λ1-ρ1(wi))≤(∑‍k:k~iρ1(wik))·(∑‍k:k~jρ1(wjk)),
that is,
(35)λ12-λ1(ρ1(wj)+ρ1(wi))+ρ1(wi)ρ1(wj)-(∑‍k:k~iρ1(wik))·(∑‍k:k~jρ1(wjk))≤0,
that is,
(36)λ1≤ρ1(wi)+ρ1(wj)2+(ρ1(wi)-ρ1(wj))2+4(∑‍k:k~iρ1(wik))(∑‍k:k~jρ1(wjk))2≤maxi~j{+(ρ1(wi)-ρ1(wj))2+4(∑‍k:k~iρ1(wik))(∑‍k:k~jρ1(wjk))2ρ1(wi)+ρ1(wj)2mmmmmm+(ρ1(wi)-ρ1(wj))2+4(∑‍k:k~iρ1(wik))(∑‍k:k~jρ1(wjk))2}.
This completes the proof of (14).
Now suppose that equality holds in (14). Then all inequalities in the previous argument must be equalities.
From equality in (23), we get
(37)x-iTx-i=x-kTx-kforallk,k~i.
Sincex-i≠0, we get that x-k≠0 for all k, k~i. From equality in (22) and Lemma 2, we get that x-i is an eigenvector of wik for the largest eigenvalue ρ1(wik). Hence we say that x-k=ax-i for some a, for any k, k~i.
On the other hand, from (37) we get
(38)a2x-iTx-i=x-iTx-i,
that is,
(39)a2=1asx-iTx-i>0.
From equality in (21), we have
(40)-∑‍k:k~ix-iTwikx-k=∑‍k:k~i|x-iTwikx-k|.
Sincex-k=ax-i, from (40) we get
(41)-∑‍ak:k~ix-iTwikx-i=∑‍k:k~i|a||x-iTwikx-k|=∑‍k:k~i|a|x-iTwikx-kasx-iTwikx-k>0.
Hence we get
(42)a=-1
from equalities in (41). Therefore we have
(43)x-k=-x-iforallk,k~i.
Similarly from equality in (29), we get that x-j is an eigenvector of wjk for the largest eigenvalue ρ1(wjk). Hence we say that x-k=bx-j for some b, for any k, k~j. From equality in (16) we have
(44)x-jTx-j=x-kTx-kfork~i,
that is,
(45)b2x-jTx-j=x-jTx-j,
that is,
(46)b2=1asx-jTx-j>0.
Applying the same methods as previously, we get(47)b=-1.
Therefore we have(48)x-k=-x-jforallk,k~j.
Fori~j(49)x-i=-x-j.
Hence we take thatU={k:x-k=x-i} and W={k:x-k=-x-i} from (43), (48), and (49). So, Nj⊂U and Ni⊂W. Also, U≠W≠∅ since x-i≠0. Further, for any vertex s∈NNi there exists a vertex r∈Ni such that r~jℓr~s, where NNi is the neighbor of neighbor set of vertex i. Therefore x-r=-x-i and x-s=x-i. So NNi⊂U. By similar argument we can present that NNj⊂W. Continuing the procedure, it is easy to see, since G is connected, that V=U∪W and that the subgraphs induced by U and W, respectively, are empty graphs. Hence G is bipartite. Moreover, x-i is a common eigenvector of wik and wi for the largest eigenvalue ρ1(wik) and ρ1(wi).
Fori,k∈U(50)λ1x-i=wix-i+∑‍k:k~iwikx-i=wkx-i+∑‍k:k~iwikx-i,
that is,
(51)wix-i=wkx-i.
Sincex-i is an eigenvector of wi corresponding to the largest eigenvalue of ρ1(wi) for all i, we get
(52)ρ1(wi)x-i=ρ1(wk)x-i,
that is,
(53)(ρ1(wi)-ρ1(wk))x-i=0,
that is,
(54)ρ1(wi)=ρ1(wk)asx-i≠0.
Therefore we get thatρ1(wi) is constant for all i∈U. Similarly we can show that ρ1(wj) is constant for all j∈W.
HenceG is a bipartite semiregular graph.
Conversely, suppose that conditions (i)-(ii) of the theorem hold for the graphG. Let G be (ρ1(wi),ρ1(wj))-semiregular bipartite graph. Let x be a common eigenvector of wik corresponding to the largest eigenvalue ρ1(wik) for all i,k. Then we have
(55)ρ1(wi)=∑‍k:k~iρ1(wik),ρ1(wj)=∑‍k:k~jρ1(wjk).
By Lemma3, we get
(56)λ1=ρ1(wi)+ρ1(wj),
that is,
(57)λ1=ρ1(wi)+ρ1(wj)2c+(ρ1(wi)-ρ1(wj))2+4(∑‍k:k~iρ1(wik))(∑‍k:k~jρ1(wjk))2.Corollary 13 (see [1]).
LetG be a simple connected weighted graph where each edge weight wi,j is a positive number. Then
(58)λ1≤maxi~j{wi+wj}.
Moreover equality holds in (58) if and only if G is bipartite semiregular graph.Proof.
We haveρ1(wi)=wi and ρ1(wij)=wij for all i, j. From Theorem 12, we get the required result.Corollary 14 (see [5]).
LetG be a simple connected unweighted graph. Then
(59)λ1≤maxi~j{di+dj},
where di is the degree of vertex i. Moreover equality holds in (59) if and only if G is a bipartite regular graph or G is a bipartite semiregular graph.Proof.
For unweighted graph,wi,j=1 for i~j. Therefore wi=di. Using Corollary 6, we get the required results.Theorem 15.
LetG be a simple connected weighted graph. Then
(60)λ1≤maxi~j{(ρ1(wi)+∑‍k:k~iρ1(wik))(ρ1(wj)+∑‍k:k~jρ1(wjk))},
where wij is the positive definite weight matrix of order p of the edge ij. Moreover equality holds in (60) if and only if(i)
G is a weighted-regular bipartite graph;(ii)
wij have a common eigenvector corresponding to the largest eigenvalue ρ1(wij) for all i, j.Proof.
LetX-=(x-1,x-2,…,x-n)T be an eigenvector corresponding to the largest eigenvalue λ1 of L(G). We assume that x-i is the vector component of X- such that
(61)x-iTx-i=maxk∈V{x-kTx-k}.
Since X- is nonzero, so is x-i. Let
(62)x-jTx-j=maxk:k~j{x-kTx-k}
be. We have
(63)L(G)X-=λ1X-.
From theith equation of (43), we have
(64)λ1x-i=wix-i-∑‍k:k~iwikx-k,
that is,
(65)λ1x-iTx-i=|x-iTwix-i-∑‍k:k~ix-iTwikx-k|≤|x-iTwix-i|+∑‍k:k~i|x-iTwikx-k|≤ρ1(wi)x-iTx-i+∑‍k:k~iρ1(wik)x-iTx-ix-kTx-kby(2)≤ρ1(wi)x-iTx-i+∑‍k:k~iρ1(wik)x-iTx-iby(40).
Hence we get
(66)λ1≤ρ1(wi)+∑‍k:k~iρ1(wik).
By the same method, from the jth equation of (43), we have
(67)λ1x-j=wjx-j-∑‍k:k~jwjkx-k,
that is,
(68)λ1x-jTx-j=|x-jTwjx-j-∑‍k:k~jx-jTwjkx-k|≤|x-jTwjx-j|+∑‍k:k~j|x-jTwjkx-k|≤ρ1(wj)x-jTx-j+∑‍k:k~jρ1(wjk)x-jTx-jx-kTx-kby(2)≤ρ1(wj)x-jTx-j+∑‍k:k~jρ1(wjk)x-jTx-jby(41).
Hence we get(69)λ1≤ρ1(wj)+∑‍k:k~jρ1(wjk).
From (49) and (58), we have
(70)λ12≤(ρ1(wi)+∑‍k:k~iρ1(wik))(ρ1(wj)+∑‍k:k~jρ1(wjk)),
that is,
(71)λ1≤maxi~j{(ρ1(wi)+∑‍k:k~iρ1(wik))(ρ1(wj)+∑‍k:k~jρ1(wjk))}.
This completes the proof of (60).
Now we show the case of equality in (60). By similar method in Theorem 12. In the part of equalit, the necessary condition can show easily. So we will show the sufficient condition.
Suppose that conditions (i)-(ii) of Theorem hold for the graphG. We must prove that
(72)λ1=maxi~j{(ρ1(wi)+∑‍k:k~iρ1(wik))(ρ1(wj)+∑‍k:k~jρ1(wjk))}.
LetG be regular bipartite graph. Therefore we have ρ1(wi)=α for i∈U and ρ1(wj)=α for j∈W such that V=U∪W. Let x- be a common eigenvector of wik corresponding to the largest eigenvalue ρ1(wik) for all i,k. Hence we have
(73)ρ1(wi)=∑‍k:k~iρ1(wik).
From (71) we get that
(74)λ1≤2α.
On the other hand, the following equation can be easily verified:(75)(2α)(x-x-⋮x--x--x-⋮-x-)cvbccc=(w1·0-w1,k+1·-w1,n0·0-w2,k+1·-w2,n…·……·…0·wk-wk,k+1·-wk,n-wk+1,1·-wk+1,kwk+1·0-wk+1,2·-wk+2,k0·0…·……·…-wn,1·-wn,k0·wn)cccbbbsd×(x-x-⋮x--x--x-⋮-x-).
Thus2α is an eigenvalue of L(G). Since λ1 is the largest eigenvalue of L(G), we get
(76)2α≤λ1.
So from (74) and (76) we obtain
(77)λ1=maxi~j{(ρ1(wi)+∑‍k:k~iρ1(wik))(ρ1(wj)+∑‍k:k~jρ1(wjk))}.Corollary 16.
LetG be a simple connected weighted graph where each edge weight wi,j is a positive number. Then
(78)λ1≤maxi~j{2wiwj}.
Moreover equality holds in (78) if and only if G is bipartite semiregular graph.Proof.
We haveρ1(wi)=wi and ρ1(wij)=wij for all i, j. From Theorem 15 we get the required result.Corollary 17.
LetG be a simple connected unweighted graph. Then
(79)λ1≤maxi~j{2didj},
where di is the degree of vertex i. Moreover equality holds in (79) if and only if G is a bipartite regular graph or G is a bipartite semiregular graph.Proof.
For unweighted graph,wi,j=1 for i~j. Therefore wi=di. Using Corollary 16, we get the required results.Example 18.
LetG1=(V1,E1) and G2=(V2,E2) be a weighted graph where V1={1,2,3,4}, E1={{1,3},{2,4},{3,4}} and each weight is the positive definite matrix of order three. Let V2={1,2,3,4,5,6,7}, E2={{1,4},{2,4},{3,4},{4,5},{5,6},{5,7}} such that each weight is the positive definite matrix of order two. Assume that the following Laplacian matrices of G1 and G2 are as follows:(80)L(G1)=[100000-1000000530000-5-30000330000-3-30000002-10000-210000-12-10001-210000-1200001-2-10000072-262-20-5-3000211126-20-3-3000-2113-2-2100002-1062-281-2000-12-126-2-38-30000-12-2-210-2-212],L(G2)=[110000-1-1000000120000-1-2000000001100-1-1000000001300-1-3000000000011-1-1000000000014-1-4000000-1-1-1-1-1-144-1-10000-1-2-1-3-1-4414-1-50000000000-1-133-1-1-1-1000000-1-5318-1-6-1700000000-1-1110000000000-1-6160000000000-1-1001100000000-1-70017].The largest eigenvalues ofL(G1) and L(G2) are λ1=25,66, λ2=26.16 rounded two decimal places and the previously mentioned bounds give the following results:(81)(6)(7)(10)(12)(14)(60)G132.9032.8827.9029.5530.8830.93G234.1229.8627.1127.2234.0533.90.ForG1, we see that the upper bounds in (14) and (60) are better than upper bounds in (6) and (7). But they are not better than upper bounds in (10) and (12) from (81).For alsoG2, we see that upper bounds in (14) and (60) are only better than the upper bound in (6).Consequently, we cannot exactly compare all the bounds for weighted graphs, where the weights are positive definite matrices. Modifications according to each weight of edges, especially for matrices can be shown.



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