On the relationship between (Af)α-spectral radii of graphs with starlike branch tree or bouquet branch graph and its linearly order

For a graph G and a vertex v of G, let $G_v(n_1,n_2,\dots ,n_d)$ be the graph obtained from G by linking the paths on $n_1,n_2,\dots ,n_d$ vertices to the vertex v of G, respectively. We denote by $d_G\left(v_i\right)$ (or $d_i$ for short) the degree of the vertex $v_i$ in G. Let $f(x,y)>0$ be a real symmetric function in x and y. The function-weighted adjacency matrix $A^f\left(G\right)$ of a graph G is a square matrix, where the $(i,j)$-entry is equal to $f(d_i,d_j)$ if the vertices $v_i$ and $v_j$ are adjacent and 0 otherwise, in which $d_i$ is the degree of the vertex $v_i$. In [22], Shan and Liu showed that the $A_{\alpha }$-spectral radius of $G_v(n_1,n_2,\dots ,n_d)$ will increase according to the shortlex ordering of $(n_1,n_2,\dots ,n_d)$. However, we find some mistakes in their proof. In this paper, we will correct their proof, and moreover, extend their results from the $A_{\alpha }$-spectral radius to the ${\left(A^f\right)}_{\alpha }$-spectral radius. In addition, let $G_v^c(n_1,n_2,\dots ,n_d)$ be the graph obtained from G by identifying a vertex from each of the cycles on $n_1,n_2,\dots ,n_d$ vertices and the vertex v of G, respectively. We will show that the ${\left(A^f\right)}_{\alpha }$-spectral radius of $G_v^c(n_1,n_2,\dots ,n_d)$ will decrease according to the majorization ordering of $(n_1,n_2,\dots ,n_d)$.