Trees with extremal spectral radius of weighted adjacency matrices among trees weighted by degree-based indices

By Xueliang Li, Zhiqian Wang. Linear Algebra and its Applications, 2021

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Abstract For a graph G = ( V , E ) and i , j ∈ V , denote by d i , d j the degrees of vertices i, j in G. Let f ( d i , d j ) > 0 be a function symmetric in i and j. Define a matrix A f ( G ) , called the weighted adjacency matrix of G, with the ij-entry A f ( G ) ( i , j ) = f ( d i , d j ) if i ∼ j and A f ( G ) ( i , j ) = 0 otherwise. In this paper, we find the extremal trees with the largest radius of A f when f ( x , y ) is increasing and convex in variable x. We also find the extremal tree with the smallest radius of A f when f ( x , y ) has a form P ( x , y ) or P ( x , y ) , where P ( x , y ) is a symmetric polynomial with nonnegative coefficients and zero constant term. This paper tries to unify the spectral study of weighted adjacency matrices of graphs weighted by some topological indices.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Organic Chemistry

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