The spectral closeness of a graph $G$ is defined as the spectral radius of the closeness matrix of $G$, whose $(u,v)$-entry for vertex $u$ and vertex $v$ is $2^{-d_G(u,v)}$ if $u\not= v$ and $0$ otherwise, where $d_G(u,v)$ is the distance between $u$ and $v$ in $G$. The residual spectral closeness of $G$ is defined as the minimum spectral closeness of the subgraphs of $G$ with one vertex deleted. We propose local grafting operations that  decrease or increase  the  spectral closeness and  determine those graphs that uniquely minimize and/or  maximize the  spectral closeness in some families of  graphs. We also discuss extremal properties of the residual spectral closeness.

Let [Formula: see text] be a simple graph of order [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text]. The arithmetic–geometric matrix [Formula: see text] of [Formula: see text] is a matrix of order [Formula: see text] defined by [Formula: see text] if [Formula: see text] and 0 otherwise, where [Formula: see text] stands for the degree of the vertex [Formula: see text] in [Formula: see text]. The arithmetic–geometric characteristic polynomial of [Formula: see text] is the characteristic polynomial of [Formula: see text]. The arithmetic–geometric energy [Formula: see text] of [Formula: see text] is the sum of absolute values of all eigenvalues of [Formula: see text]. In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted.

For a graph $G$ with vertex set $V(G)$ and $u,v\in V(G)$, the distance between vertices $u$ and $v$ in $G$, denoted by $d_G(u,v)$, is the length of a shortest path connecting them and it is $\infty$ if there is no such a path, and the closeness of vertex $u$ in $G$ is $c_G(u)=\sum_{w\in V(G)}2^{-d_G(u,w)}$. Given a graph $G$ that is not necessarily connected, for $u,v\in V(G)$, the closeness matrix of $G$ is the matrix whose $(u,v)$-entry is equal to $2^{-d_G(u,v)}$ if $u\ne v$ and $0$ otherwise, the closeness Laplacian is the matrix whose $(u,v)$-entry is equal to \[ \begin{cases} -2^{-d_G(u,v)} & \mbox{if } u\ne v,\\ c_G(u) & \mbox{otherwise} \end{cases} \] and the closeness signless Laplacian  is the matrix whose $(u,v)$-entry is equal to  \[ \begin{cases} 2^{-d_G(u,v)} & \mbox{if } u\ne v,\\ c_G(u) & \mbox{otherwise}. \end{cases} \] We establish relations  connecting the spectral properties of closeness Laplacian and  closeness signless Laplacian and structural properties of graphs. We give tight upper  bounds for all nontrivial closeness Laplacian  eigenvalues  and characterize the extremal graphs, and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian  eigenvalue. Also, we give tight upper  bounds for the  closeness signless Laplacian  eigenvalues and determine the trees whose largest closeness signless Laplacian  eigenvalues achieve  the first two largest values.

The Pareto H-eigenvalues of nonnegative tensors and (adjacency tensors of) uniform hypergraphs are studied. Particularly, it is shown that the Pareto H-eigenvalues of a nonnegative tensor are just the spectral radii of its weakly irreducible principal subtensors, and those hypergraphs that minimize or maximize the second largest Pareto H-eigenvalue over several well-known classes of uniform hypergraphs are determined.