On the Normalized Laplacian Energy(Randi c Energy) Ay se Dilek - PowerPoint PPT Presentation

Some Fundamental Definitions Preliminary Results Main Results References On the Normalized Laplacian Energy(Randi´ c Energy) Ay¸ se Dilek Maden Sel¸ cuk University, Konya/Turkey aysedilekmaden@selcuk.edu.tr SGA 2016- Spectral Graph Theory and Applications May 18-20, 2016 Belgrade, SERBIA Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Outline 1 Some Fundamental Definitions 2 Preliminary Results 3 Main Results 4 References Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions In this talk, we report some bounds for the normalized Laplacian energy and Randi´ c energy of a connected (bipartite) graph. Firstly, we give some fundamental definitions which are used in our results. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions In this talk, we report some bounds for the normalized Laplacian energy and Randi´ c energy of a connected (bipartite) graph. Firstly, we give some fundamental definitions which are used in our results. Let G be undirected and simple graph with | V ( G ) | = n vertices and | E ( G ) | = m edges. Furthermore, for i = 1 , 2 , · · · , n , the degree of a vertex v i in V ( G ) will be denoted by d i . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions In this talk, we report some bounds for the normalized Laplacian energy and Randi´ c energy of a connected (bipartite) graph. Firstly, we give some fundamental definitions which are used in our results. Let G be undirected and simple graph with | V ( G ) | = n vertices and | E ( G ) | = m edges. Furthermore, for i = 1 , 2 , · · · , n , the degree of a vertex v i in V ( G ) will be denoted by d i . If any vertices v i and v j are adjacent, then we use the notation v i ∼ v j . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions It is known that we also have the Laplacian matrix related to the adjacency and diagonal matrices. In fact, for a diagonal matrix D ( G ) whose ( i , i )-entry is d i , the Laplacian matrix L ( G ) of G is defined as L ( G ) = D ( G ) − A ( G ). Since A ( G ) and L ( G ) are all real symmetric matrices, their eigenvalues are real numbers. So we assume that λ 1 ( G ) ≥ λ 2 ( G ) ≥ · · · ≥ λ n − 1 ( G ) ≥ λ n ( G ) ( µ 1 ( G ) ≥ µ 2 ( G ) ≥ · · · ≥ µ n − 1 ( G ) ≥ µ n ( G )) are the adjaceny (Laplacian) eigenvalues of G . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions Because the graph G is assumed to be connected, it has no isolated vertices and therefore the matrix D ( G ) − 1 / 2 is well defined. Then L ∗ = L ∗ ( G ) = D ( G ) − 1 / 2 L ( G ) D ( G ) − 1 / 2 is called the normalized Laplacian matrix of the graph G . Its eigenvalues are ρ 1 ( G ) ≥ ρ 2 ( G ) ≥ · · · ≥ ρ n − 1 ( G ) ≥ ρ n ( G ) . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References It is convenient to write the normalized Laplacian matrix as I n − R , where R is the so-called Randi´ c matrix , whose ( i , j )-entry is 1 � √ , if v i ∼ v j d i d j r ij = 0 , otherwise [ Maden et al.-2010 ] The Randi´ c eigenvalues q 1 ( G ), q 2 ( G ),..., q n ( G ) of the graph G are the eigenvalues of its Randi´ c matrix. Since R is real symmetric matrix, its eigenvalues are real number. So we can order them so that q 1 ( G ) ≥ q 2 ( G ) ≥ · · · ≥ q n ( G ) . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions M -energy of G is n � � � λ i ( M ) − tr ( M ) � � � E M ( G ) = � , � � n i =1 where tr ( M ) is the trace of M . The energy of a graph was introduced by Gutman in 1978 as n � E ( G ) = | λ i ( G ) | . i =1 Recently, the adjacency enery, Laplacian energy, Randi´ c energy and normalized Laplacian energy of a graph has received much interest. Along the some lines, the energy of more general matrices and sequences has been studied. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions Using the above equality with M taken to be L ∗ , the normalized Laplacian energy and Randi´ c energy of a graph G is n n � � E L ∗ ( G ) = | ρ i − 1 | andE R ( G ) = | q i | , i =1 i =1 respectively. Since L ∗ = I n − R , it easy to see that this is equivalent to n � E L ∗ ( G ) = | q i | = E R ( G ) . i =1 In the literature, some basic properties of E L ∗ ( G ) may be found. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions Now, recall that the Randi´ c index of a graph G is defined as Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Some Fundamental Definitions Now, recall that the Randi´ c index of a graph G is defined as � ( d i d j ) α , R α = R α ( G ) = v i ∼ v j where the summation is over all edges v i v j in G , and α � = 0 is a fixed real number. The general Randi´ c index when α = − 1 is 1 � R − 1 = R − 1 ( G ) = , d i d j v i ∼ v j Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Preliminary Results Now, we recall some results from spectral graph theory and state a few analytical spectral inequalities for our work. Lemma (2.2) [ F. Chung -1997 ] Let the normalized Laplacian eigenvalues of G be given as ρ 1 ≥ ρ 2 ≥ · · · ≥ ρ n = 0 . Then 0 ≤ ρ i ≤ 2 . Morover ρ 1 = 2 if and only if G has a connected bipartite nontrivial component. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Preliminary Results Lemma (2.3) [ P. Zumstein -2005 ] Let G be a graph with n vertices and normalized Laplacian matrix L ∗ without isolated vertices. Then n � ρ i = n i =1 and n ρ i 2 = n + 2 R − 1 . � i =1 Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Preliminary Results Lemma (2.4) [ L. Shi -2009 ] Let G be a graph of order n with no isolated vertices. Suppose that G has minimum verwerte degree equal to d min and maximum vertex degree equal to d max . Then n n ≤ R − 1 ≤ 2 d max 2 d min Equality occurs in both bounds if and only if G is a regular graph. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Main Results After all above materials, we are ready to present our main results. The following results are also valid for Randi´ c energy. Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Main Results Theorem (3.1) Let G be undirected , simple and connected graph with n,n ≥ 3 vertices . Then � 2 n − 1 ≤ E L ∗ ( G ) = E R ( G ) 1 + 2 R − 1 + ( n − 1)( n − 2)∆ � 2 ≤ 1 + ( n − 2)(2 R − 1 − 1) + ( n − 1)∆ (1) n − 1 where ∆ = det ( I n − L ∗ ) . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Main Results Remark In [ Hakimi-Nezhaad et al.-2014 ], Hakimi-Nezhaad et al. obtained the following lower bound for the normalized Laplacian energy : � n � n − 1 � 2 n − 1 . E L ∗ ( G ) ≥ 1 + − 1 + 2 ∆ (2) d max 2 From Lemma (2 . 4) , the lower bound (1) is better than (2) . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)

Some Fundamental Definitions Preliminary Results Main Results References Main Results Considering Lemma (2 . 4) and the inequality (1), we arrive at the following result. Corollary Let G be a graph of order n with no isolated vertices. Suppose that G has minimum vertex degree equal to d min and maximum vertex degree equal to d max . Then � n 2 n − 1 ≤ E L ∗ ( G ) = E R ( G ) 1 + − 1 + ( n − 1)( n − 2)∆ d max � n � � 2 ≤ 1 + ( n − 2) − 1 + ( n − 1)∆ (3) n − 1 d min where ∆ = det ( I n − L ∗ ) . Ay¸ se Dilek Maden On the Normalized Laplacian Energy(Randi´ c Energy)