In \(1989\), Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\) and they obtained sufficient conditions for a graph to be hamiltonian with the implicit degrees. In this paper, we prove that if \(G\) is a \(2\)-connected graph of order \(n\) with \(\alpha(G) \leq n/2\) such that \(\text{id}(v) \geq (n-1)/2\) for each vertex \(v\) of \(G\), then \(G\) is hamiltonian with some exceptions.

The compact, Fredholm, and isometric weighted composition operators are characterized in this paper.

We discuss the chromaticity of one family of \(K_4\)-homeomorphs with exactly two non-adjacent paths of length two, where the other four paths are of length greater than or equal to three. We also give a sufficient and necessary condition for the graphs in the family to be chromatically unique.

In this paper, we deduced the following new Stirling series:

\[ n! \sim \sqrt{2n\pi} (\frac{n}{2})^n exp(\frac{1}{12n+1}[1 + \frac{1}{12n} (1+\frac{\frac{2}{5}}{n} + \frac{\frac{29}{150}}{n^2} – \frac{\frac{62}{2625}}{n^3} – \frac{\frac{9173}{157500}}{n^4} +\ldots )^{-1}]) ,\]

which is faster than the classical Stirling’s series.

For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:

\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]

A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).

We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in \([11]\) if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.

The general neighbor-distinguishing total chromatic number \(\chi”_{gnd}(G)\) of a graph \(G\) is the smallest integer \(k\) such that the vertices and edges of \(G\) can be colored by \(k\) colors so that no adjacent vertices have the same set of colors. It is proved in this note that \(\chi”_{gnd}(G) = \lceil \log_2 \chi(G) \rceil + 1\), where \(\chi(G)\) is the vertex chromatic number of \(G\).

A sequence \(A\) is a \(B_h^*[g]\) sequence if the coefficients of \((\sum_{a\in A}(z)^a)^h\) are bounded by \(g\). The standard Sidon sequence is a \(B[2]\) sequence. Finite Sidon sequences are called Golomb rulers, which are found to have many applications such as error correcting codes, radio frequency selection, and radio antennae placement. Let \(R_h(g,n)\) be the largest cardinality of a \(B[g]\) sequence contained in \(\{1,2,\ldots,n\}\), and \(F(h,g,k) = \min\{n : R_h(g,n) \geq k\}\). In this paper, computational techniques are applied to construct optimal generalized Sidon sequences, and \( 49\) new exact values of \(F(2,g,k)\) are found.

Recently, Chu \([5]\) derived two families of terminating \(_2F_1(2)\)-series identities. Their \(q\)-analogues will be established in this paper.

Let \(H\), \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \(G-GD_\lambda(H)\), is a partition of all the edges of \(H\) into subgraphs (called \(G\)-blocks), each of which is isomorphic to \(G\). A large set of \(G-GD_\lambda(H)\), denoted by \(G-LGD_\lambda(H)\), is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \(G-GD_\lambda(H)\)s. In this paper, we determine the existence spectrums for \(K_{2,2}-LGD_\lambda(K_{m,n})\).

A binary vertex coloring (labeling) \(f: V(G) \to \mathbb{Z}_2\) of a graph \(G\) is said to be friendly if the number of vertices labeled 0 is almost the same as the number of vertices labeled 1. This friendly labeling induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(uv) = f(u)f(v)\) for all \(uv \in E(G)\). Let \(e_f(i) = |\{uv \in E(G) : f^*(uv) = i\}|\) be the number of edges of \(G\) that are labeled \(i\). The product-cordial index of the labeling \(f\) is the number \(pc(f) = |e_f(0) – e_f(1)|\). The product-cordial set of the graph \(G\), denoted by \(PC(G)\), is defined by

\[PC(G) = \{pc(f): f \text{ is a friendly labeling of } G\}.\]

In this paper, we will determine the product-cordial sets of long grids \(P_m \times P_n\), introduce a class of fully product-cordial trees and suggest new research directions in this topic.