A bound on the spectral radius of a weighted graph

Let G be simple, connected weighted graphs, where the edge weights are positive definite matrices. In this paper, we will give an upper bound on the spectral radius of the adjacency matrix for a graph G and characterize graphs for which the bound is attained

Estimating the spectral radius of a graph by the second Zagreb index

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Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees

AbstractLet G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by μ(G)=μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0. A vertex of degree one is called a pendant vertex. Let Tn,k be a tree with n vertices, which is obtained by adding paths P1,P2,…,Pk of almost equal the number of its vertices to the pendant vertices of the star K1,k. In this paper, the following results are given:(1) Let T be a tree with n vertices and k pendant vertices. Thenμ(T)⩽μ(Tn,k),where equality holds if and only if T is isomorphic to Tn,k.(2) Let G be a simple connected bipartite graph with degrees d1,d2,…,dn. Thenμ(G)⩾21n∑i=1ndi2,where equality holds if and only if G is a regular connected bipartite graph.(3) Let G be a simple connected bipartite graph with vertices v1,v2,…,vn and their degrees d1,d2,…,dn. Thenμ(G)⩾2+1m∑vi∼vj,i<j(di+dj-2)2,where m is the edge number of G and equality holds if and only if G is either a regular connected bipartite graph or a semiregular connected bipartite graph or the path with four vertices

Some relations among the largest eigenvalues of product matrix and graph matrices

( ) ve ( ) sırasıyla bir grafının komşuluk matrisi ve nokta derecelerinin bir köşegen matrisi olmak üzere ( ) ( ) ( ) matrisini tanımlarız. Bu makalede bu çarpım matrisinin spektral yarıçapı ile graf matrislerinin en büyük öz değerleri arasında bazı bağıntılar elde edilmiştir. Ayrıca nümerik sonuçlar da verilmiştir.We define product matrix as ( ) ( ) ( ) , where ( ) is an adjacency matrix and ( ) is a diagonal matrix of vertex degrees of a graph . In this paper, some relations among the spectral radius of product matrix and the largest eigenvalues of graph matrices are obtained. We also give numerical results for them