Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Let G = (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 weightmatrices 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. Let G be a weighted graph on n vertices. Denote by w i,j the positive definite weight matrix of order p of the edge ij, and assume that w ij = w ji . We write i ∼ j if vertices i and j are adjacent. Let w i = ∑ j:j∼i w ij . be the weight matrix of the vertex i. The Laplacian matrix of a graph G is defined as L(G) =


Introduction
Let  = (, ) be simple graphs, as graphs which have no loops or parallel edges such that  is a finite set of vertices and  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.
Let  be a weighted graph on  vertices.Denote by  , the positive definite weight matrix of order  of the edge , and assume that   =   .We write  ∼  if vertices  and  are adjacent.Let   = ∑ :∼   .be the weight matrix of the vertex .
The Laplacian matrix of a graph  is defined as () = (  ), where otherwise. ( The zero denotes the  ×  zero matrix.Hence () is square matrix of order .Let  1 denote the largest eigenvalue of ().In this paper we also use to avoid the confusion that  1 (  ) is the spectral radius of   matrix.If  is the disjoint union of two nonempty sets  1 and  2 such that every vertex  in  1 has the same  1 (  ) and every vertex  in  2 has the same  1 (  ), then  is called a weight-semiregular graph.If  1 (  ) =  1 (  ) in weight semiregular graph, then  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][2][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][2][3].We also compare the upper bounds with the known upper bounds in [1][2][3].We also characterize graphs which achieve the upper bound.The results clearly generalize some known results for weighted and unweighted graphs.

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.
Lemma 2 (Horn and Johnson [4]).Let  be a Hermitian × matrix with eigenvalues Equality holds if and only if  is an eigenvector of  corresponding to  1 and  =  for some  ∈ .
Lemma 3 (see [1]).Let  be a ( ) . ( Theorem 4 (see [1]).Let G be a simple connected weighted graph.Then where   is the positive definite weight matrix of order p of the edge .Moreover equality holds in (6) if and only if (i)  is a weight-semiregular bipartite graph, (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Theorem 5 (see [2]).Let G be a simple connected weighted graph.Then where   is the positive definite weight matrix of order p of the edge .Moreover equality holds in (7) if and only if (i)  is a bipartite semiregular graph; (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Corollary 6 (see [2]).Let  be a simple connected weighted graph where each edge weight   is a positive number.Then where   = (∑ :∼     )/  and   is the weight of vertex .Moreover equality holds if and only if  is a bipartite regular graph.
Corollary 7 (see [2]).Let  be a simple connected weighted graph where each edge weight   is a positive number.Then where   = (∑ :∼     )/  and   is the weight of vertex .Moreover equality holds if and only if  is a bipartite semiregular graph.
Theorem 8 (see [2]).Let G be a simple connected weighted graph.Then where   = (∑ :∼  1 (  ) 1 (  ))/ 1 (  ) and   is the positive definite weight matrix of order p of the edge .Moreover equality holds in (10) if and only if (i)  is a weighted-regular graph or  is a weight-semiregular bipartite graph; (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Corollary 9 (see [2]).Let  be a simple connected weighted graph where each edge weight   is a positive number.Then where   = (∑ :∼     )/  and   is the weight of vertex .Moreover equality holds if and only if  is a bipartite semiregular graph or  is a bipartite regular graph.
Theorem 10 (see [3]).Let G be a simple connected weighted graph.Then where   is the positive definite weight matrix of order  of the edge  and   ∩   is the set of common neighbours of  and .Moreover equality holds in (12) if and only if (i)  is a weight-semiregular bipartite graph; (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Corollary 11 (see [3]).Let  be a simple connected weighted graph where each edge weight   is a positive number.Then Moreover equality holds if and only if  is a bipartite semiregular graph.

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. Let G be a simple connected weighted graph.Then where   is the positive definite weight matrix of order p of the edge .Moreover equality holds in (14) if and only if (i)  is a weighted-regular graph or  is a weightsemiregular bipartite graph; (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Proof.Let  = ( 1 ,  2 , . . .,   )  be an eigenvector corresponding to the largest eigenvalue  1 of ().We assume that   is the vector component of  such that Since  is nonzero, so is   .Let We have From the th equation of (18), we have that is, ≤ ∑ From (23) we have International Journal of Combinatorics that is, From the th equation of (18), we get that is, Similarly, from (30) we get ≤ ∑ that is, So, from (25) and (32) we have Hence we get that is, that is, 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 Since   ̸ = 0, we get that   ̸ = 0 for all ,  ∼ .From equality in (22) and Lemma 2, we get that   is an eigenvector of   for the largest eigenvalue  1 (  ).Hence we say that   =   for some , for any ,  ∼ .
On the other hand, from (37) we get that is, (41) Hence we get from equalities in (41).Therefore we have Similarly from equality in (29), we get that   is an eigenvector of   for the largest eigenvalue  1 (  ).Hence we say that   =   for some , for any ,  ∼ .From equality in (16) we have that is, that is, Applying the same methods as previously, we get Therefore we have For  ∼ Hence we take that  = { :   =   } and  = { :   = −  } from ( 43), (48), and (49).So,   ⊂  and   ⊂ .Also,  ̸ =  ̸ = 0 since   ̸ = 0. Further, for any vertex  ∈    there exists a vertex  ∈   such that  ∼ ℓ ∼ , where    is the neighbor of neighbor set of vertex .Therefore   = −  and   =   .So    ⊂ .By similar argument we can present that    ⊂ .Continuing the procedure, it is easy to see, since  is connected, that  =  ∪  and that the subgraphs induced by  and , respectively, are empty graphs.Hence  is bipartite.Moreover,   is a common eigenvector of   and   for the largest eigenvalue  1 (  ) and  1 (  ).
For ,  ∈ that is, Since   is an eigenvector of   corresponding to the largest eigenvalue of  1 (  ) for all , we get that is, that is, Therefore we get that  1 (  ) is constant for all  ∈ .Similarly we can show that  1 (  ) is constant for all  ∈ .
Hence  is a bipartite semiregular graph.
Conversely, suppose that conditions (i)-(ii) of the theorem hold for the graph .Let  be ( 1 (  ),  1 (  ))semiregular bipartite graph.Let  be a common eigenvector of   corresponding to the largest eigenvalue  1 (  ) for all , .Then we have By Lemma 3, we get that is, (57) Corollary 13 (see [1]).Let  be a simple connected weighted graph where each edge weight  , is a positive number.Then Moreover equality holds in (58) if and only if  is bipartite semiregular graph.
Corollary 14 (see [5]).Let  be a simple connected unweighted graph.Then where   is the degree of vertex .Moreover equality holds in (59) if and only if  is a bipartite regular graph or  is a bipartite semiregular graph.
Theorem 15.Let G be a simple connected weighted graph.Then International Journal of Combinatorics where   is the positive definite weight matrix of order p of the edge .Moreover equality holds in (60) if and only if (i)  is a weighted-regular bipartite graph; (ii)   have a common eigenvector corresponding to the largest eigenvalue  1 (  ) for all , .
Proof.Let  = ( 1 ,  2 , . . .,   )  be an eigenvector corresponding to the largest eigenvalue  1 of ().We assume that   is the vector component of  such that Since  is nonzero, so is   .Let be.We have From the th equation of (43), we have Hence we get By the same method, from the th equation of (43), we have (68) Hence we get From ( 49) and (58), we have that is, 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 graph .We must prove that Let  be regular bipartite graph.Therefore we have  1 (  ) =  for  ∈  and  1 (  ) =  for  ∈  such that  =  ∪ .Let  be a common eigenvector of   corresponding to the largest eigenvalue  1 (  ) for all , .Hence we have From (71) we get that Corollary 17.Let  be a simple connected unweighted graph.Then where   is the degree of vertex .Moreover equality holds in (79) if and only if  is a bipartite regular graph or  is a bipartite semiregular graph.
For also  2 , 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.