In this paper, we investigate some interesting identities on the Euler numbers and polynomials arising from their generating functions and difference operators. Finally, we give some properties of Bernoulli and Euler polynomials by using \(p\)-adic integral on \(\mathbb{Z}_p\).

Let \(\pi\) be a finite projective plane of order \(n\). Consider the substructure \(\pi_{n+2}\) obtained from \(\pi\) by removing \(n+2\) lines (including all points on them) no three of which are concurrent. In this paper, firstly, it is shown that \(\pi_{n+2}\) is a B-L plane and it is also homogeneous. Let \(PG(3,2)\) be a finite projective \(3\)-space of order \(n\). The substructure obtained from \(PG(3,2)\) by removing a tetrahedron that is four planes of \(PG(3,n)\) no three of which are collinear is a finite hyperbolic \(3\)-space (Olgun-Ozgir [10]). Finally, we prove that any two hyperbolic planes with the same parameters are isomorphic in this hyperbolic \(3\)-space. These results appeared in the second author’s MSc thesis.

Let \(G\) be a graph of order \(n\), and let \(a\) and \(b\) be integers such that \(1 \leq a < b\). Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) < f(x) \leq b\) for each \(x \in V(G)\). Then \(G\) has a \((g, f)\)-factor if the minimum degree \(\delta(G) \geq \frac{(b-1)^2-(a+1)(a+b-1)}{a+1}\) ,\(n>\frac{(a+b)(a+b-1)}{a+1}\) and \(\max\{d_G(x), d_G(y)\} \geq \frac{(b-1)n}{a+b}\) for any two nonadjacent vertices \(x\) and \(y\) in \(G\). Furthermore, it is shown that the result in this paper is best possible in some sense.

In this note, we consider the on-line Ramsey numbers \(\overline{R}(P_n, P_m)\) for paths. Using a high-performance computing cluster, we calculated the values for off-diagonal numbers for paths of lengths at most \(8\). Also, we were able to check that \(\overline{R}(P_9, P_9) = 17\), thus solving the problem raised in [5].

In this paper, we determine the images of hyperbolic ellipses under the Möbius and harmonic Möbius transformations.

Given a disjoint union of some complete graphs, one can define a graph by choosing one vertex from each complete graph and making all of these vertices adjacent. This observation leads us to define a new operation on certain graphs. We compute the characteristic polynomial of the resulting graphs and indicate a method for computing the determinant of this matrix for obtaining the characteristic polynomial of new graphs. We show that line graphs of trees can be obtained by performing this operation on some graphs, and, as an application, we compute the characteristic polynomials of line graphs of trees.

Consider a connected undirected graph \(G = (V, E)\) and an integer \(r \geq 1\); for any vertex \(v \in V\), let \(B_r(v)\) denote the ball of radius \(r\) centred at \(v\), i.e., the set of all vertices linked to \(v\) by a path of at most \(r\) edges. If for all vertices \(v \in V\), the sets \(B_r(v)\) are different, then we say that \(G\) is \(r\)-twin-free.

In \(r\)-twin-free graphs, we prolong the study of the extremal values that can be reached by some classical parameters in graph theory, and investigate here the maximum degree.

A new construction of authentication codes with arbitration from \((2\nu-2+2+1)\)-dimensional singular pseudo-symplectic geometry on finite fields is given. Assuming that the encoding rules are chosen according to a uniform probability distribution, the parameters and the probabilities of success for different types of deceptions are also computed.

By a defensive alliance in a graph \(G\) we mean any set \(S\) of vertices in \(G\) such that each vertex in \(S\) is adjacent to at least as many vertices inside \(S\), including the vertex itself, as outside \(S\). If, in addition, we require that every vertex outside a defensive alliance \(S\) is adjacent to at least one vertex in \(S\), then \(S\) becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of \(G\). In this paper, we determine bounds for the global defensive alliance numbers in the join, corona, and composition of graphs.

Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). A triangle is a cycle of length three. As usual, \(K_n\) denotes the complete graph on \(n\) vertices. It is shown that for all nonnegative integers \(p\) and \(q\) and for all positive integers \(n\), \(K_n\) can be decomposed into \(p\) copies of \(P_4\) and \(q\) copies of \(C_3\) if and only if \(3(p+q) = e(K_n)\), \(p \neq 1\) if \(n\) is odd, and \(p \geq \frac{n}{2}\) if \(n\) is even.