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.

The aim of this note is to present a short proof of a result of Alaeiyan et al. [Bull. Austral. Math. Soc.\( 77 (2008) 315-323;\)
Proc. Indian Acad. Sci., Math. Sci. \(119 (2009) 647-653\)] concerning the non-existence of cubic semisymmetric graphs of order \(8p\) or \(8p^2\), where \(p\) is a prime. In those two papers, the authors choose the heavy weaponry of covering techniques. Our proof relies on the analysis of the subgroup structure of the full automorphism group of the graph and the normal quotient graph theory.

Let \(G\) be a graph of order \(n\) with adjacency matrix \(A(G)\) and diagonal degree matrix \(D(G)\). The generalized characteristic polynomial of \(G\) is defined to be \(f_G(x,t) = \det (xI_n – (A(G) – tD(G)))\). The \(R\)-graph of \(G\), denoted by \(R(G)\), is obtained by adding a new vertex for each edge of \(G\) and joining each new vertex to both end vertices of the corresponding edge. The generalized \(R\)-vertex corona, denoted by \(R(G) \boxdot \wedge _i^n H\), is the graph obtained from \(R(G)\) and \(H\) by joining the \(i\)-th vertex of \(V(G)\) to every vertex of \(H\). In this paper, we determine the generalized characteristic polynomial of \(R(G) \boxdot \wedge _i^n H\). As applications, we get infinitely many pairs of generalized cospectral graphs, the number of spanning trees and Kirchhoff index of \(R(G) \boxdot\wedge _i^n H\).

In this paper, we introduce an \(O(n^2)\) time algorithm to determine the cyclic edge connectivity of a planar graph, where \(n\) is the order of the planar graph. This is the first correct square time algorithm for cyclic edge connectivity of planar graphs.