In this paper, we give a four parameter theta function identity and prove it by using some properties of Jacobi’s theta functions and Jacobi’s fundamental formulae.

The order dimension is an invariant on partially ordered sets introduced by Dushnik and Miller in \(1941 [1]\). It is known that the computation of the order dimension of a partially ordered set in general is highly complex,with current algorithms relying on the minimal coloring of an associated hypergraph, see \([5]\). The aim of this work is to extend the family of posets whose order dimension is easily determined by a formula. We introduce an operation called layering. Finally, we provide the precise formulas for determining the order dimension of any given number of layers of Trotter’s generalized crowns.

In this paper, the regular endomorphisms of a split graph are investigated. We give a condition under which the regular endomorphisms of a split graph form a monoid.

The clique graph \(K(G)\) of a graph \(G\) is the intersection graph of all its (maximal) cliques, and \(G\) is said to be clique divergent if the order of its \(n\)-th iterated clique graph \(K^n(G)\) tends to infinity with \(n\). In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form \(C_n(a, b, c)\) with \(0 < a < b < c < \frac{n}{3}\). Such a circulant is clique divergent if and only if it is not clique-Helly. Owing to the Dragan-Szwarcfiter Criterion to decide clique-Hellyness, our result implies that the clique divergence of these circulants can be decided in polynomial time. Our main difficulty was the case \(C_n(1, 2, 4)\), which is clique divergent but no previously known technique could be used to prove it.

A total dominating set \(S\) of a graph \(G\) with no isolated vertex is a locating-total dominating set of \(G\) if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\) are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian product of cycles \(C_m\) and \(C_n\), and prove that for any positive integer \(n \geq 3\), the locating-total domination numbers of the Cartesian product of cycles \(C_3\) and \(C_n\) is equal to \(n\) for \(n \equiv 0 \pmod{6}\) or \(n + 1\) otherwise.

A graph \(G\) is called a fractional \((g, f, m)\)-deleted graph if after deleting any \(m\) edges, then the resulting graph admits a fractional \((g, f)\)-factor. In this paper, we prove that if \(G\) is a graph of order \(n\), and if \(1 \leq g(x) \leq f(x) \leq 6\) for any \(x \in V(G)\), \(\delta(G) \geq \frac{b^2(i-1)}{a} ++2m\), \(n > \frac{(a+b)(i(a+b)+2m-2)}{a}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_i)| \geq \frac{bn}{a+b} \), for any independent set \(\{x_1, x_2, \ldots,x_i\}\) of \(V(G)\), where \(i \geq 2\), then \(G\) is a fractional \((g, f, m)\)-deleted graph. The result is tight on the neighborhood union condition.

In this short paper, we introduce the second order linear recurrence relation of the \(AB\)-generalized Fibonacci sequence and give the explicit formulas for the sums of the positively and negatively subscripted terms of the \(AB\)-generalized Fibonacci sequence by matrix methods. This sum generalizes the one obtained earlier by Kilig in \([2]\).

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.