Open Journal of Discrete Applied Mathematics (ODAM)

Let \(G = (X, Y; E)\) be a bipartite graph with two vertex partition subsets \(X\) and \(Y\). \(G\) is said to be balanced if \(|X| = |Y|\) and \(G\) is said to be bipancyclic if it contains cycles of every even length from \(4\) to \(|V(G)|\). In this note, we present spectral conditions for the bipancyclic bipartite graphs.

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify some particular sums such as the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive formulae for simplifying general repeated sums as well as a variant containing binomial coefficients. Additionally, we study the \(m\)-th difference of a sequence and show how sequences whose \(m\)-th difference is constant can be related to binomial coefficients.

We interact the theory of possibility Pythagorean bipolar fuzzy soft sets, possibility bipolar fuzzy soft sets and define complementation, union, intersection, AND and OR. The possibility Pythagorean bipolar fuzzy soft sets are presented as a generalization of soft sets. Notably, we tend to showed De Morgan’s laws, associate laws and distributive laws that are holds in possibility Pythagorean bipolar fuzzy soft set theory. Also, we advocate an algorithm to solve the decision making problem primarily based on soft set model.

A novel vertex-degree-based topological invariant, called Nirmala index, was recently put forward, defined as the sum of the terms \(\sqrt{d(u)+d(v)}\) over all edges \(uv\) of the underlying graph, where \(d(u)\) is the degree of the vertex \(u\). Based on this index, we now introduce the respective “Nirmala matrix”, and consider its spectrum and energy. An interesting finding is that some spectral properties of the Nirmala matrix, including its energy, are related to the first Zagreb index.

In this work, we investigate the initial boundary-value problem for a parabolic type Kirchhoff equation with logarithmic nonlinearity. We get the existence of global weak solution, by the potential wells method and energy method. Also, we get results of the decay and finite time blow up of the weak solutions.

This furthers the notions of parametric equivalence, isomorphism and uniqueness in graphs. Results for certain cycle related graphs are presented. Avenues for further research are also suggested.

The Kolakoski sequence $S$ is the unique element of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) starting with 1 and coinciding with its own run length encoding. We use the parity of the lengths of particular subclasses of initial words of \(S\) as a unifying tool to address the links between the main open questions – recurrence, mirror/reversal invariance and asymptotic density of digits. In particular we prove that recurrence implies reversal invariance, and give sufficient conditions which would imply that the density of 1s is \(\frac{1}{2}\).

The energy of a graph is the sum of absolute values of its eigenvalues. The nullity of a graph is the algebraic multiplicity of number zero in its spectrum. Empirical facts indicate that graph energy decreases with increasing nullity, but proving this property is difficult. In this paper, a method is elaborated by means of which the effect of nullity on graph energy can be quantitatively estimated.

This short paper introduces the notions of parametric equivalence, isomorphism and uniqueness in graphs. Results for paths, cycles and certain categories (or types) of trees with regards to minimum confluence sets are presented.

A variation of Dyck paths allows for down-steps of arbitrary length, not just one. Credits for this invention are given to Emeric Deutsch. Surprisingly, the enumeration of them is somewhat akin to the analysis of Motzkin-paths; the last section contains a bijection.