We determine all connected odd graceful graphs of order \(\leq 6\). We show that if \(G\) is an odd graceful graph, then \(G \cup K_{m,n}\) is odd graceful for all \(m, n \geq 1\). We give an analogous statement to the graceful graphs statement, and we show that some families of graphs are odd graceful.

In this paper, we provide a method to obtain the lower bound on the number of distinct maximum genus embeddings of the complete bipartite graph \(K_{n,n}\) (\(n\) is an odd number), which, in some sense, improves the results of S. Stahl and H. Ren.

For positive integer \(n\), let \(f_3(n)\) be the least upper bound of the sums of the lengths of the sides of \(n\) cubes packed into a unit cube \(C\) in three dimensions in such a way that the smaller cubes have sides parallel to those of \(C\). In this paper, we improve the lower bound of \(f_3(n)\).

The transformation graph \(G^{+- -}\) of a graph \(G\) is the graph with vertex set \(V(G) \cup E(G)\), in which two vertices \(u\) and \(uv\) are joined by an edge if one of the following conditions holds: (i) \(u,v \in V(G)\) and they are adjacent in \(G\), (ii) \(u,v \in E(G)\) and they are not adjacent in \(G\), (iii) one of \(u\) and \(wv\) is in \(V(G)\) while the other is in \(E(G)\), and they are not incident in \(G\). In this paper, for any graph \(G\), we determine the independence number and the connectivity of \(G^{+- -}\). Furthermore, we show that for a graph \(G\) with no isolated vertices, \(G^{+- -}\) is hamiltonian if and only if \(G\) is not a star and \(G \not\in \{2K_2, K_2\}\).

We introduce quasi-almostmedian graphs as a natural nonbipartite generalization of almostmedian graphs. They are filling a gap between quasi-median graphs and quasi-semimedian graphs. We generalize some results of almostmedian graphs and deduce some results from a bigger class of quasi-semimedian graphs. The consequence of this is another characterization of almostmedian graphs as well as two new characterizations of quasi-median graphs.

In this note, we establish a convolution formula for Bernoulli polynomials in a new and brief way, and some known results are derived as a special case.

In this study, we define the generalized \(k\)-order Fibonacci matrix and the \(n \times n\) generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. We find the inverse of the generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. In the last section, we factorize this matrix via the generalized \(k\)-order Fibonacci matrix and give illustrative examples for these factorizations.

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let \(\mathcal{G}\) be the set of unicyclic graphs of order \(n\) with girth \(g\). For all integers \(n\) and \(g\) with \(5 \leq g \leq n – 6\), we determine the first \(|\frac{g}{2}| + 3\) spectral radii of unicyclic graphs in the set \(\mathcal{U}_n^g\).

In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using \(\{0,1,2,\ldots,k-1\}\) is called \(k\)-equitable if the number of vertices (resp. edges) labeled \(i\) and the number of vertices (resp. edges) labeled \(j\) differ by at most one and is called \(k\)-balanced if the number of vertices labeled \(i\) and the number of edges labeled \(j\) differ by at most one. We determine which graphs in certain families are \(k\)-equitable or \(k\)-balanced and we give also some necessary conditions on these two labelings.