The maximum possible toughness among graphs with \(n\) vertices and \(m\) edges is considered for \(m \geq \lceil n^2/4 \rceil\). We thus extend results known for \(m \geq n\lfloor n/3 \rfloor\). When \(n\) is even, all of the values are determined. When \(n\) is odd, some values are determined, and the difficulties are discussed, leaving open questions.

In this paper, we, by means of Rosa’s \(\alpha\)-labelling and \(k\)-graceful labelling, prove that generalized spiders, generalized caterpillars, and generalized path-block chains are graceful under some conditions. Some of the results are stronger than that obtained in \([4]\).

We study convexity with respect to a definition of fractional independence in a graph \(G\) that is quantified over neighbourhoods rather than edges. The graphs that admit a so-called universal maximal fractional independent set are characterized, as are all such sets. A characterization is given of the maximal fractional independent sets which cannot be obtained as a proper convex combination of two other such sets.

In this paper, we consider the relationship between the toughness and the existence of fractional \(f\)-factors. It is proved that a graph $G$ has a fractional \(f\)-factor if \(t(G) \geq \frac{b^2+b}{a}-\frac{b+1}{b}\). Furthermore, we show that the result is best possible in some sense.

It is always fascinating to see what results when seemingly different areas of mathematics overlap. This article reveals one such result; number theory and linear algebra are intertwined to yield complex factorizations of the classic Fibonacci, Pell, Jacobsthal, and Mersenne numbers. Also, in this paper we define a new matrix generalization of the Fibonacci numbers, and using essentially a matrix approach we show some properties of this matrix sequence.

The notion of meandric polygons is introduced in this paper. A bijection exists between the set of meandric polygons and that of closed meanders. We use these polygons to enumerate the set of meanders which have a fixed number of arcs of the meandric curves lying above and below the horizontal line at a given point.

In this paper, we give a sufficient and necessary condition for a \(k\)-extendable graph to be \(2k\)-factor-critical when \(k = \frac{v}{4}\), and prove some results on independence numbers in \(n\)-factor-critical graphs and \(k\frac{1}{2}\)-extendable graphs.

In this paper, we show that some families of graphs are arbitrarily graceful or almost graceful.

We consider the lattice of order ideals of the union of an \(n\)-element fence and an antichain of size \(i\), whose Hasse diagram turns out to be isomorphic to the \(i\)-th extended Fibonacci cube. We prove that the Whitney numbers of these lattices form a unimodal sequence satisfying a particular property, called \({alternating}\), we find the maximum level of the game sequence and determine the exact values of these numbers.