In this paper we give upper bounds of the number of edges in four types of labeled graphs of known orders.

In this paper, some lattices generated by the orbits of the subspaces under finite classical groups are considered. the characteristic polynomials of these lattices are obtained by using the effective approach by Aigner in \([2]\) , and their expressions are also determined.

In this paper, we study \(CT\)-burst array error \([6]\) detection and correction in row-cyclic array codes \([8]\).

In this work, we define a new integer sequence related to Fibonacci and Pell sequences with four parameters and then derived some algebraic identities on it including, the sum of first non-zero terms, recurrence relations, rank of its terms, powers of companion matrix and the limit of cross-ratio of four consecutive terms of it.

The hexagonal system considered here, denoted by \({E}_n^2\), is formed by \(3n\) (\(n \geq 2\)) hexagons shown in Fig. 2(a). In this paper, we give the explicit expression of the characteristic polynomial \(\Phi_A({E}_n^2, x)\). Subsequently, we obtain the multiplicity of eigenvalues \(+1\), the spectral radius, and the nullity of \({E}_n^2\). Furthermore, the energy, Estrada index, and the number of Kekulé structures of \({E}_n^2\) are determined.

The frequency assignment problem originated in researching mobile communication networks. A proper total coloring of a graph \(G\) is a coloring of both edges and vertices of \(G\) such that no two adjacent or incident elements receive the same color. As known, the vertex distinguishing total coloring is one of the suitable tools for investigating the frequency assignment problem. We introduce a new graph total coloring, called \((4)\)-adjacent vertex distinguishing total coloring (\((4)\)-AVDTC), in this paper. Our coloring contains the adjacent vertex distinguishing total coloring. The minimum number of colors required for every \((4)\)-AVDTC of \(G\) is called the \((4)\)-AVDTC chromatic number of \(G\). We will show that using at most \(\Delta(G) + 4\) colors can achieve at least \(4\) different adjacent vertex distinguishing actions for some communication networks \(G\). The exact \((4)\)-AVDTC chromatic numbers of several classes of graphs are determined here and a problem is presented.

Let \(R\) be a commutative ring with identity and \(T(\Gamma(R))\) its total graph. The subject of this article is the investigation of the properties of the corresponding line graph \(L(T(\Gamma(R)))\). The classification of all commutative rings whose line graphs are planar or toroidal is given. It is shown that for every integer \(g \geq 0\) there are only finitely many commutative rings such that \(\gamma(L(T(\Gamma(R)))) = g\).

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An \(n \times 7\) array based on \(\{0, 1, \ldots, nd\}\) is called a sparse anti-magic square of order \(n\) with density \(d\) (\(d < n\)), denoted by SAMS\((n, d)\), if its row-sums, column-sums, and two main diagonal sums constitute a set of \(2n + 2\) consecutive integers. A SAMS\((n, d)\) is called regular if there are \(d\) positive entries in each row, each column, and each main diagonal. In this paper, some constructions of regular sparse anti-magic squares are provided and it is shown that there exists a regular SAMS\((n, d-1)\) if and only if \(n \geq 4\).

In this paper, we perform a further investigation for the \(q\)-analogues of the classical Bernoulli numbers and polynomials. By applying summation transform techniques, we establish some new recurrence relations for these type numbers and polynomials. We also present some illustrative special cases as well as immediate consequences of the main results.

The toughness \(t(G)\) of a noncomplete graph \(G\) is defined as \[t(G) = \min\left\{\frac{|S|}{w(G – S)} \mid S \subset V(G), w(G – S) \geq 2\right\}\] and the toughness of a complete graph is \(\infty\), where \(w(G – S)\) is the number of connected components of \(G – S\). In this paper, we give the sharp upper and lower bounds for the Kronecker product of a complete graph and a tree. Moreover, we determine the toughness of the Kronecker product of a complete graph and a star, a path, respectively.