We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_p(G)=(\sum _{uv\in E(G)}|d(u)-d(v){|}^p{)}^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in $G$. The new index is not only generalization of the well-known Albertson irregularity index and $\sigma$-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.

Let $G$ be a simple connected graph with the vertex set $V(G)$ and $d_B(u,v)$ be the biharmonic distance between two vertices $u$ and $v$ in $G$. The biharmonic index $BH(G)$ of $G$ is defined as

$$BH(G)=\frac{1}{2}\sum _{u\in V(G)}\sum _{v\in V(G)}{d}_B^2(u,v)=n\sum _{i=2}^n\frac{1}{{\lambda }_i^2(G)},$$

where ${\lambda }_i(G)$ is the $i$-th eigenvalue of the Laplacian matrix of $G$ with $n$ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index and the Kirchhoff index. In addition, the extremal value on the biharmonic index for all graphs with diameter two, trees and firefly graphs are given, respectively. Finally, some graph operations on the biharmonic index are presented.

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. For any real number $\alpha \in [0,1]$, Nikiforov recently defined the $A_{\alpha }$-matrix of $G$ as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$. The graph invariant $S_{\alpha }^p(G)$ is the sum of the $p$-th power of the $A_{\alpha }$-eigenvalues of $G$ for $\frac{1}{2}<\alpha <1$, which has a close relation to the $\alpha$-Estrada index. In this paper, we establish some bounds on $S_{\alpha }^p(G)$ and characterize the extremal graphs. In particular, we present some bounds on $S_{\alpha }^p(G)$ in terms of the degree sequences, order and size of $G$ by using majorization techniques. Moreover, we give lower and upper bounds for $S_{\alpha }^p(G)$ of a bipartite graph and characterize the extremal graphs.

From geometric point of view, we introduced the Sombor-Wiener index of a graph and studied the basic properties of the new index. It was shown that the Sombor-Wiener index was useful in predicting the acentric factor of octane isomers. In addition, we proposed a degree-weighted Wiener index to generalize the Schultz index, the Gutman index, and the Sombor-Wiener index. Meanwhile, we gave the calculation formula of degree-weighted Wiener index for generalized Bethe trees.