A vertex-colored path is vertex-rainbow if its internal vertices have distinct colors. For a connected graph $G$ with connectivity $\kappa (G)$ and an integer $k$ with $1\le k\le \kappa (G)$, the rainbow vertex $k$-connectivity of $G$ is the minimum number of colors required to color the vertices of $G$ such that any two vertices of $G$ are connected by $k$ internally vertex-disjoint vertex-rainbow paths. In this paper, we determine the rainbow vertex $k$-connectivities of all small cubic graphs of order $8$ or less.

For a simple graph $G=(V,E)$, a vertex labeling $\alpha :V\to \{1,2,\dots ,k\}$ is called a $k$-labeling. The weight of an edge $xy$ in $G$, denoted by $w_{\varphi }(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{\varphi }(xy)=\varphi (x)+\varphi (y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$ there is $w_{\varphi }(e)\ne w_{\varphi }(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, denoted by $\mathrm{e}\mathrm{s}(G)$. In this paper, we determine the exact value for certain families of graphs with path $P_2$.

We give a $q$-analogue of some Dixon-like summation formulas obtained by Gould and Quaintance [Fibonacci Quart. 48 (2010), 56-61] and Chu [Integral Transforms Spec. Funct. 23 (2012), 251-261], respectively. For example, we prove that
$\sum _{k=0}^{2m}(-1{)}^{m-k}{q}^{(\genfrac{}{}{0ex}{}{m-k}{2})}(\genfrac{}{}{0ex}{}{2m}{k})(\genfrac{}{}{0ex}{}{x+k}{2m+r})(\genfrac{}{}{0ex}{}{x+2m-k}{2m+r})$ = $\frac{q^{m(x-m-r)}(\genfrac{}{}{0ex}{}{2m}{m})}{(\genfrac{}{}{0ex}{}{2m+r}{m})}(\genfrac{}{}{0ex}{}{x}{m+r})(\genfrac{}{}{0ex}{}{x+m}{m+r})$ where $(\genfrac{}{}{0ex}{}{x}{k})$ denotes the $q$-binomial coefficient.

A pentangulation is a simple plane graph such that each face is bounded by a cycle of length $5$. We consider two diagonal transformations in pentangulations, called $\mathcal{A}$ and $\mathcal{B}$. In this paper, we shall prove that any two pentangulations with the same number of vertices can be transformed into each other by $\mathcal{A}$ and $\mathcal{B}$. In particular, if they are not isomorphic to a special pentangulation, then we do not need $\mathcal{B}$.

The harmonic index $H(G)$ of a graph $G$ is defined as the sum of the weights of all edges $uv$ of $G$, where the weight of $uv$ is $\frac{2}{d(u)+d(v)}$, with $d(u)$ denoting the degree of the vertex $u$ in $G$. In this work, we compute the harmonic index of a graph with a cut-vertex and with more than one cut-vertex. As an application, this topological index is computed for Bethe trees and dendrimer trees. Also, the harmonic indices of Fasciagraph and a special type of trees, namely, polytree, are computed.

Let $G^{\sigma }$ be an oriented graph obtained by assigning an orientation $\sigma$ to the edge set of a simple undirected graph $G$. Let $S(G^{\sigma })$ be the skew adjacency matrix of $G^{\sigma }$. The skew energy of $G^{\sigma }$ is defined as the sum of the absolute values of all eigenvalues of $S(G^{\sigma })$. In this paper, we give the skew energy order of a family of digraphs and determine the oriented bicyclic graphs of order $n\ge 13$ with the first five largest skew energies, which extends the results of the paper [X. Shen, Y. Hou, C. Zhang, Bicyclic digraphs with extremal skew energy, Electron. J. Linear Algebra 23 (2012) 340-355].

Let $P_n$ denote the $n$-th Catalan-Larcombe-French number. Recently, the $2$-log-convexity of the Catalan-Larcombe-French sequence was proved by Sun and Wu. Moreover, they also conjectured that the quotient sequence $\{\frac{P_n}{P_{n-1}}{\}}_{n=0}^{\mathrm{\infty }}$ of the Catalan-Larcombe-French sequence is log-concave. In this paper, this conjecture is confirmed by utilizing the upper and lower bounds for $\frac{P_n}{P_{n-1}}$ and finding a middle function $f(n)$.

It is claimed in [13] that the metric dimension of the Möbius ladder $M_n$ is $3$ when $n\not\equiv 2(\mathrm{mod}8)$, but it is wrong; we give a counterexample when $n\equiv 6(\mathrm{mod}8)$. In this paper, we not only give the correct metric dimension in this case but also solve the open problem regarding the metric dimension of $M_n$ when $n\equiv 2(\mathrm{mod}8)$. Moreover, we conclude that $M_n$ has two subfamilies with constant metric dimensions.

An edge-colored graph $G$ is (strong) rainbow connected if any two vertices are connected by a (geodesic) path whose edges have distinct colors. The (strong) rainbow connection number of a connected graph $G$, denoted by $\mathrm{s}\mathrm{r}\mathrm{c}(G)$ (resp. $\mathrm{r}\mathrm{c}(G)$), is the smallest number of colors that are needed in order to make $G$ (strong) rainbow connected. The join $P_m\vee P_n$ of $P_m$ and $P_n$ is the graph consisting of $P_m\cup P_n$, and all edges between every vertex of $P_m$ and every vertex of $P_n$, where $P_m$ (resp. $P_n$) is a path of $m$ (resp. $n$) vertices. In this paper, the precise values of $\mathrm{r}\mathrm{c}(P_m\vee P_n)$ and $\mathrm{s}\mathrm{r}\mathrm{c}(P_m\vee P_n)$ are given for any positive integers $m$ and $n$.

Let $MG(i,n)$ be a connected molecular graph without multiple edges on $n$vertices whose minimum degree of vertices is $i$, where $i\le i\le 4$. One of the newest topological indices is the first Geometric-Arithmetic index. In this paper, we determine the graph with the minimum and the maximum value of the first Geometric-Arithmetic index in the family of graphs $MG(i,n)$,$l\le i\le 3$.