In this paper two authentication codes with multiple arbiters are constructed to protect the communication system against the attacks from the opponent, transmitter, receiver and dishonest arbiters. The first construction takes advantage of set theory to give an authentication codes with two arbiters that resists collusion attacks from dishonest arbiters and participators availably. The second construction makes full use of of Reed- Solomon-code (\(RS\)-code) and \((k, n)\)-threshold scheme to give an authentication codes with \(n\) arbiters that effectively prevents multiple arbiters from cheating.

A directed Toeplitz graph is a digraph with a Toeplitz adjacency matrix. In this paper we contribute to [6]. The paper [6] investigates the hamiltonicity of the directed Toeplitz graphs \(T_n\langle s_1,s_2,…, s_k;t_1, t_2,…,t_l\rangle\) with \(s_2 = 2\) and in particular those with \(s_3 = 3\). In this paper we extend this investigation to \(s_2 = 3\) with \(s_1 =t_1 =1\).

W. Y. C. Chen and R. P. Stanley have characterized the symmetries of the \(n\)-cube that act as derangements on the set of \(k\)-faces. In this paper we aim to use their result to characterize those finite subgroups of symmetries whose non-trivial members are derangements of the set of \(k\)-faces.

A sequential labeling of a simple graph G (non-tree) with m edges is an injective labeling f such that the vertex labels \(f(x)\) are from \({0,1,…,m-1}\) and the edge labels induced by \(f(x) + f(y)\) for each edge \(xy\) are distinct consecutive positive integers. A graph is sequential if it has a sequential labeling. We give some properties of sequential labeling and the criterion to verify sequential labeling. Necessary and sufficient conditions are obtained for every case of sequential graphs. A complete characterization of non-tree sequential graphs is obtained by vertex closure. Also, characterizations of sequential trees are given. The structure of sequential graphs is revealed.

In this paper, we give explicit algorithms to compute generating functions of some special sequences, based on the operations of differential operators and shift operators in the non-commutative context and Zeilberger’s holonomic algorithm.
It can be found that not only ordinary generating functions and exponential generating functions but also generating functions of the general form \(\sum_{n} a_n(x)w(y, n)\) can now be computed automatically. Moreover, we generalize this approach and present explicit algorithms to compute \(2\)-variable ordinary power series generating functions and mixed-type generating functions. As applications, various examples are given in the paper.

The graphs we consider are all countable. A graph \(U\) is universal in a given set \(\mathcal{P}\) of graphs if every graph in \(\mathcal{P}\) is an induced subgraph of \(U\) and \(U \in \mathcal{P}\). In this paper we show the existence of a universal graph in the set of all countable graphs with block order bounded by a fixed positive integer. We also investigate some classes of interval graphs and work towards finding universal graphs for them. The sets of graphs we consider are all examples of induced-hereditary graph properties.

In this paper, we give the Hahn polynomials represents by Carlitz’s \(q\)-operators, then show how to deduce Carlitz type generating functions by the technique of exponential operator decomposition.

The Wiener polarity index of a graph \(G\), denoted by \(W_p(G)\), is the number of unordered pairs of vertices \(u, v\) such that the distance between \(u\) and \(v\) is three, introduced by Harold Wiener in 1947. This index is utilized to demonstrate quantitative structure-property relationships in various acyclic and cyclic hydrocarbons. In this paper, we investigate the Wiener polarity index on the Cartesian, direct, strong, and lexicographic products of two non-trivial connected graphs.

Given a graph \(G := (V, E)\) and an integer \(k \geq 2\), the \({component \;order\; edge connectivity}\) of \(G\) is the smallest size of an edge set \(D\) such that the subgraph induced by \(G – D\) has all components of order less than \(k\). Let \({G}(n,m)\) denote the collection of simple graphs \(G\) with \(n\) vertices and \(m\) edges. In this paper, we investigate properties of component order edge connectivity for \({G}(n,m)\), particularly proving results on the maximum and minimum values of this connectivity measure for \({G}(n,m)\) specific values of \(n\), \(m\), and \(k\).

Let \(D\) be a simple digraph without loops and parallel arcs. Deng and Kelmans [A. Deng, A. Kelmans, Spectra of digraph transformations, Linear Algebra and its Applications, \(439(2013) 106-132]\) gave the definition of transformation digraphs by introducing symbol \(‘0’\) and \(‘1’\), and investigated the regular and spectra of digraph transformation. In this paper we discuss a class of total transformation digraphs associate with symbol \(‘0’\). Furthermore, we determine the regularity of these ten new kinds of total transformation digraphs and also give necessary and sufficient conditions for them to be strongly connected.