For a simple graph \(G = (V, E)\), a vertex labeling \(\alpha: V \rightarrow \{1, 2, \ldots, k\}\) is called a \(k\)-labeling. The weight of an edge \(xy\) in \(G\), denoted by \(w_\phi(xy)\), is the sum of the labels of end vertices \(x\) and \(y\), i.e., \(w_\phi(xy) = \phi(x) + \phi(y)\). 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 \(w_\phi(e) \neq w_\phi(f)\). The minimum \(k\) for which the graph \(G\) has an edge irregular \(k\)-labeling is called the edge irregularity strength of \(G\), denoted by \(\mathrm{es}(G)\). 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
\(\sum\limits_{k=0}^{2m} (-1)^{m-k} q^{\binom{m-k}{2}} \binom{2m} {k} \binom{x+k} {2m+r}\binom{x+2m-k} {2m+r}\) = \(\frac{q^{m(x-m-r)}\binom{2m}{m}}{\binom{2m+r}{m}}\binom{x}{m+r}\binom{x+m}{m+r}\) where \(\binom{x}{k}\) 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 \(n \geq 13\) 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 \(\{\frac{P_{n}}{P_{n-1}}\}_{n= 0}^\infty\) 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 \(n \not\equiv 2 \pmod{8}\), but it is wrong; we give a counterexample when \(n \equiv 6 \pmod{8}\). 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 \(n \equiv 2 \pmod{8}\). 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 \(\mathrm{src}(G)\) (resp. \(\mathrm{rc}(G)\)), 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 \(\mathrm{rc}(P_m \vee P_n)\) and \(\mathrm{src}(P_m \vee P_n)\) 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 \(i \leq i \leq 4\). 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 \(M{G}(i,n)\),\(l\leq i \leq 3\).

Two graphs are said to be Tutte-equivalent if their Tutte polynomials are equal. In this paper, we provide several different constructions for Tutte-equivalent graphs, including some that are not self-complementary but Tutte-equivalent to their complements (the Akiyama-Harary problem) and some “large” Tutte-equivalent graphs obtained from “small” Tutte-equivalent graphs by \(2\)-sum operations.