Some bounds on the largest eigenvalue of degree-based weighted adjacency matrix of a graph.

Let f (x , y) > 0 be a real symmetric function. For a connected graph G , the weight of edge v i v j is equal to the value f (d i , d j) , where d i is the degree of vertex v i. The degree-based weighted adjacency matrix is defined as A f (G) , in which the (i , j) -entry is equal to f (d i , d j) if v i v j is an edge of G and 0 otherwise. In this paper, we first give some bounds of the weighted adjacency eigenvalue λ 1 (A f (G)) in terms of λ 1 (A f (H)) , where H is obtained from G by some kinds of graph operations, including deleting vertices, deleting an edge and subdividing an edge, and examples are given to show that bounds are tight. Second, we obtain some bounds for the largest weighted adjacency eigenvalue λ 1 (A f (G)) of irregular weighted graphs. [ABSTRACT FROM AUTHOR]