Only few results concerning crossing numbers of join of some graphs are known. In the paper, for the special graph \(G\) on six vertices, we give the crossing numbers of \(G\vee P_n\) and \(G\vee C_n\), \(P_n\) and \(C_n\) are the path and cycle on \(n\) vertices, respectively.

Recently, Dere and Simsek have treated some applications of umbral algebra. related to several special polynomials(see \([8]\)). In this paper, we derive some new and interesting identities of special polynomials involving Bernoulli, Euler and Laguerre polynomials arising from umbral calculus.

In this paper, we prove that for any tree \(T\), \(T^2\) is a divisor graph if and only if \(T\) is a caterpillar and the diameter of \(T\) is less than six. For any caterpillar \(T\) and a positive integer \(k \geq 1\) with \(diam(T) \leq 2k\), we show that \(T^k\) is a divisor graph. Moreover, for a caterpillar \(T\) and \(k \geq 3\) with \(diam(T) = 2k\) or \(diam(T) = 2k + 1\), we show that \(T^k\) is a divisor graph if and only if the centers of \(T\) have degree two.

To construct a large graph from two smaller ones that have same order, one can add an arbitrary perfect matching between their vertex-sets. The topologies of many networks are special cases of these graphs. An interesting and important problem is how to persist or even improve their link reliability and link fault-tolerance. Traditionally, this may be done by optimizing the edge connectivity of their topologies, a more accurate method is to improve their \(m\)-restricted edge connectivity. This work presents schemes for optimizing \(m\)- restricted edge connectivity of these graphs, some well-known results are direct consequences of our observations.

In this paper we introduce a new kind of generalized Pell numbers. This generalization is introduced in the distance sense. We give different interpretations and representations of these numbers.We present relations between distance Pell numbers and Fibonacci numbers. Moreover we describe graph interpretations of distance Pell numbers. These graphs interpretations in the natural way imply a new kind of generalized Jacobsthal numbers.

A graph \(G\) is called a fractional \((g, f, n’, m)\)-critical deleted graph if after deleting any \(n’\) vertices of \(G\) the remaining graph is a fractional \((g, f, m)\)-deleted graph. In this paper, we give two binding number conditions for a graph to be a fractional \((g, f, n’, m)\)-critical deleted graph.

In this paper, we compute the hyper-Wiener index of arbitrary \(k\)-membered ring spiro chain. We also determine the extremal \(k\)-membered ring spiro chains for hyper-Wiener index.

In this paper, the notion of cyclic bursts in array codes equipped with a non-Hamming metric \([13]\) as a generalization of classical cyclic bursts \([5]\) is introduced and some bounds are obtained on the parameters of array codes for the detection and correction of cyclic burst array errors.

Let \(G\) be a graph, and let \(a\), \(b\), \(k\) be integers with \(0 \leq a \leq b\), \(k \geq 0\). An \([a, b]\)-factor of graph \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(F)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\) the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, it is proved that, if \(a\), \(b\), \(k\) be integers with \(1 \leq a < b\), \(k \geq 0\) and \(b \geq a(k+1)\) and \(G\) is a graph with \(\delta(G) \geq a+k\) and binding number \(b(G) \geq a-1+\frac{a(k+1)}{b}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.