In this study, by using Jacobsthal and Jacobsthal Lucas matrix sequences, we define \(k\)-Jacobsthal and \(k\)-Jacobsthal Lucas matrix sequences depending on one parameter \(k\). After that, by using two parameters \((s,t)\), we define \((s,t)\)-Jacobsthal and \((s,t)\)-Jacobsthal Lucas matrix sequences. And then, we establish combinatoric representations of all of these matrices.

A graph \(G\) is \(1\)-planar if it can be embedded in the plane \(\mathbb{R}^2\) so that each edge of \(G\) is crossed by at most one other edge. In this paper, we show that each \(1\)-planar graph of maximum degree \(\Delta\) at least \(7\) with neither intersecting triangles nor chordal \(5\)-cycles admits a proper edge coloring with \(\Delta\) colors.

Dirac showed that in a \((k-1)\)-connected graph there is a path through each \(k\) vertices. The path \(k\)-connectivity \(\pi_k(G)\) of a graph \(G\), which is a generalization of Dirac’s notion, was introduced by Hager in 1986. Recently, Mao introduced the concept of path \(k\)-edge-connectivity \(\omega_k(G)\) of a graph \(G\). Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\omega_4(G \circ H) \geq \omega_4(G) |V(H)|\) for any two graphs \(G\) and \(H\). Moreover, the bound is sharp.

A graph \(G = (V(G), E(G))\) is even graceful and equivalently graceful, if there exists an injection \(f\) from the set of vertices \(V(G)\) to \(\{0, 1, 2, 3, 4, \ldots, 2|E(G)|\}\) such that when each edge \(uv\) is assigned the label \(|f(u) – f(v)|\), the resulting edge labels are \(2, 4, 6, \ldots, 2|E(G)|\). In this work, we use even graceful labeling to give a new proof for necessary and sufficient conditions for the gracefulness of the cycle graph. We extend this technique to odd graceful and super Fibonacci graceful labelings of cycle graphs via some number theoretic concept, called a balanced set of natural numbers.

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every \(1\)-planar graph without \(4\)-cycles or adjacent \(5\)-vertices is \(5\)-colorable.

In previous researches on classification problems, there are some similar results obtained between \(f\)-coloring and \(g_c\)-coloring. In this article, the author shows that there always are coincident classification results for a regular simple graph \(G\) when the \(f\)-core and the \(g_c\)-core of \(G\) are same and \(f(v) = g(v)\) for each vertex \(v\) in the \(f\)-core (the \(g_c\)-core) of \(G\). However, it is not always coincident for nonregular simple graphs under the same conditions. In addition, the author obtains some new results on the classification problem of \(f\)-colorings for regular graphs. Based on the coincident correlation mentioned above, new results on the classification problem of \(g_c\)-colorings for regular graphs are deduced.

The integrity of a graph \(G = (V, E)\) is defined as \(I(G) = \min\{|S| + m(G-S): S \subseteq V(G)\}\), where \(m(G-S)\) denotes the order of the largest component in the graph \(G-S\). This is a better parameter to measure the stability of a network, as it takes into account both the amount of work done to damage the network and how badly the network is damaged. Computationally, it belongs to the class of intractable problems known as NP-hard. In this paper, we develop a heuristic algorithm to determine the integrity of a graph. Extensive computational experience on \(88\) randomly generated graphs ranging from \(20\%\) to \(90\%\) densities and from \(100\) to \(200\) vertices has shown that the proposed algorithm is very effective.

The half of an infinite lower triangular matrix \(G = (g_{n,k})_{n,k\geq 0}\) is defined to be the infinite lower triangular matrix \(G^{(1)} = (g^{(1)}_{n,k \geq 0})\) such that \(g^{(1)}_{n,k} = g_{2n-k,n}\) for all \(n \geq k \geq 0\). In this paper, we will show that if \(G\) is a Riordan array, then its half \(G^{(1)}\) is also a Riordan array. We use Lagrange inversion theorem to characterize the generating functions of \(G^{(1)}\) in terms of the generating functions of \(G\). Consequently, a tight relation between \(G^{(1)}\) and the initial array \(G\) is given, hence it is possible to invert the process and rebuild the original Riordan array \(G\) from the array \(G^{(1)}\). If the process of taking half of a Riordan array \(G\) is iterated \(r\) times, then we obtain a Riordan array \(G^{(r)}\). The further relation between the result array \(G^{(r)}\) and the initial array \(G\) is also considered. Some examples and applications are presented.

A graph \(G\) is list \(k\)-arborable if for any sets \(L(v)\) of cardinality at least \(k\) at its vertices, one can choose an element (color) for each vertex \(v\) from its list \(L(v)\) so that the subgraph induced by every color class is an acyclic graph (a forest). In the paper, it is proved that every planar graph with \(5\)-cycles not adjacent to \(3\)-cycles and \(4\)-cycles is list \(2\)-arborable.

For two vertices \(u\) and \(v\) in a strong digraph \(D\), the strong distance between \(u\) and \(v\) is the minimum number of arcs of a strong subdigraph of \(D\) containing \(u\) and \(v\). The strong eccentricity of a vertex \(v\) of \(D\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong diameter (strong radius) of \(D\) is the maximum (minimum) strong eccentricity among all vertices of \(D\). The lower orientable strong diameter (lower orientable strong radius), \(\mathrm{sdiam}(G)\) (\(\mathrm{srad}(G)\)), of a 2-edge-connected graph \(G\) is the minimum strong diameter (minimum strong radius) over all strong orientations of \(G\). In this paper, a conjecture of Chen and Guo is disproved by proving \(\mathrm{sdiam}(K_{3} \square K_{3}) = \mathrm{sdiam}(K_{3} \square K_{4}) = 5\), \(\mathrm{sdiam}(K_{m} \square P_{n})\) is determined, \(\mathrm{sdiam}(G)\) and \(\mathrm{srad}(G)\) for cycle vertex multiplications are computed, and some results concerning \(\mathrm{sdiam}(G)\) are described.