A \(d\)-antimagic labeling of a plane graph \(G = (V, E, F)\) is a one-to-one mapping taking the vertices, edges, and faces onto the integers \(1, 2, \ldots, |V(G)| + |E(G)| + |F(G)|\) so that the \(s\)-sided face weights form an arithmetic progression of difference \(d\). This paper describes \(d\)-antimagie labelings for Möbius grids.

There are networks that can be modeled by simple graphs, where edges are perfectly reliable but nodes are subject to failure, e.g. hardwired computer systems. One measure of the “vulnerability” of the network is the connectivity \(\kappa\) of the graph. Another, somewhat related, vulnerability parameter is the component order connectivity \(\kappa_c^{(k)}\), i.e. the smallest number of nodes that must fail in order to ensure that all remaining components have order less than some value \(k\). In this paper we present necessary and sufficient conditions on a 4-tuple \((n,k,a,b)\) for a graph \(G\) to exist having \(n\) nodes, \(\kappa = a\), and \(\kappa_c^{(k)} = b\). Sufficiency of the conditions follows from a specific construction described in our work. Using this construction we obtain ranges of values for the number of edges in a graph having \(n\) nodes, \(\kappa = a\), and \(\kappa_c^{(k)} = b\) thereby obtaining sufficient conditions on the \(5\)-tuple \((n,e,k,a,b)\) for a graph to exist having \(n\) nodes, \(e\) edges, \(\kappa = a\), and \(\kappa_c^{(k)} = b\). In a limited number of special cases, we show the conditions on \((n,e,k,a,b)\) to be necessary as well.

In this paper, we develop a polynomial time algorithm to determine the cyclic edge connectivity of a \(k\)-regular graph for \(k \geq 3\). The time complexity of the algorithm is bounded by \(O(k^{11}|V|^8)\), in particular, it is \(O(|V|^8)\) for cubic graphs.

For each integer \(m \geq 1\), consider the graph \(G_m\) whose vertex set is the set \(\mathbb{N} = \{0,1,2,\ldots\}\) of natural numbers and whose edges are the pairs \(xy\) with \(y = x+m\), \(y = x-m\), \(y = mx\), or \(y = \frac{x}{m}\). Our aim in this note is to show that, for each \(m\), the graph \(G_m\) contains a Hamilton path. This answers a question of Lichiardopol.

Given a partial \(K_4\)-design \((X, {P})\), if \(x \in X\) is a vertex which occurs in exactly one block of \({P}\), then call \(x\) a free vertex. In this paper, a technique is described for obtaining a cubic embedding of any partial \(K_4\)-design with the property that every block in the partial design contains at least two free vertices.

The average distance \(\mu(D)\) of a strong digraph \(D\) is the average of the distances between all ordered pairs of distinct vertices of \(D\). Plesnik \([6]\) proved that if \(D\) is a strong tournament of order \(n\), then \(\mu(D) \leq \frac{n+4}{6} + \frac{1}{n}\). In this paper, we show that if \(D\) is a \(k\)-connected tournament of order \(n\), then \(\mu(D) \leq \frac{n}{6k} + \frac{19}{6} + \frac{k}{n}\). We demonstrate that, apart from an additive constant, this bound is best possible.

A subset \(U\) of a set \(S\) with a binary operation is called avoidable if \(S\) can be partitioned into two subsets \(A\) and \(B\) such that no element of \(U\) can be written as a product of two distinct elements of \(A\) or as the product of two distinct elements of \(B\). The avoidable sets of the bicyclic inverse semigroup are classified.

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by

\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]

We call the matrix \((a_{n,m})_{n,m\geq 0}\) an generalized Seidel matrix with a parameter pair \((\alpha, \beta)\). If \(\alpha = \beta = 1\), then this matrix is the classical Seidel matrix. For various different parameter pairs \((\alpha, \beta)\) we will impose some evenness or oddness conditions on the exponential generating functions of the initial sequence \(a_{0,m}\) and the final sequence \(a_{n,0}\) of a generalized Seidel matrix (i.e., we require that these generating functions or certain related functions are even or odd). These conditions imply that the initial sequences and final sequences are equal to well-known classical sequences such as those of the Euler numbers, the Genocchi numbers, and the Springer numbers.

As applications, we give a straightforward proof of the continued fraction representations of the ordinary generating functions of the sequence of Genocchi numbers. And we also get the continued fractions representations of the ordinary generating functions of the Genocchi polynomials, Bernoulli polynomials, and Euler polynomials. Lastly, we give some applications of congruences for the Euler polynomials.

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(f: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).

In this paper, we give necessary and sufficient conditions for the cordiality of the \(t\)-ply \(P_t(u,v)\), i.e. a thread of ply number \(t\).

A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over \(\mathbb{F}_2\). Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broué-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over \(\mathbb{F}_{2f}\). We show how to construct Jacobi forms with various index over the totally real field. This is one of extension of Broué-Enguehard map.