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.

The study of chromatically unique graphs has been drawing much attention and many results are surveyed in \([4, 12, 13]\). The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu \([17]\) (see also \([4]\)). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu \([17]\) and Dong et al. \([4]\) respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs \(C_n(P_m)\), the graph obtained from a path \(P_m\) and a cycle \(C_n\) by identifying a pendant vertex of the path with a vertex of the cycle. Let \(\bar{G}\) stand for the complement of a graph \(G\). We prove the following results:

1. The graph \(\overline{{{C}_{n-1}(P_2)}}\) is chromatically unique if and only if \(n \neq 5, 7\).
2. Almost every \(\overline{{C_n(P_m)}}\) is not chromatically unique, where \(n \geq 4\) and \(m \geq 2\).

An \(L(2,1)\)-labelling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \((2,1)\)-labelling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labelling with \(\max\{f(v) : v \in V(G)\} = k\). Griggs and Yeh conjecture that \(\lambda(G) \leq \Delta^2\) for any simple graph with maximum degree \(\Delta \geq 2\). This article considers the graphs formed by the cartesian product of \(n\) (\(n \geq 2\) graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].