Graceful labeling of graphs is used in radar codes. In this work, we introduce a new version of gracefulness, which we call edge-even graceful labeling of graphs. We establish a necessary and sufficient condition for edge-even graceful labeling of path graphs \(P_n\), cycle graphs \(C_n\), and star graphs \(K_{1,n}\). We also prove some necessary and sufficient conditions for some path and cycle-related graphs, namely, Friendship, Wheel, Double wheel, and Fan graphs.

The Hamiltonian problem is a classical problem in graph theory. Most of the research on the Hamiltonian problem is looking for sufficient conditions for a graph to be Hamiltonian. For a vertex \(v\) of a graph \(G\), Zhu, Li, and Deng introduced the concept of implicit degree \(id(v)\), according to the degrees of its neighbors and the vertices at distance \(2\) with \(v\) in \(G\). In this paper, we will prove that: Let \(G\) be a \(2\)-connected graph on \(n \geq 3\) vertices. If the maximum value of the implicit degree sums of \(2\) vertices in \(S\) is more than or equal to \(n\) for each independent set \(S\) with \(\kappa(G) + 1\) vertices, then \(G\) is Hamiltonian.

Let \((d_1, d_2, \dots, d_n)\) be a sequence of positive integers with \(n-1 \geq d_1 \geq d_2 \geq \dots \geq d_n\). We give a characterization of \((d_1, d_2, \dots, d_n)\) that is the degree sequence of a graph with cyclomatic number \(k\). This simplifies the characterization of Erdős-Gallai.

We explore new combinatorial properties of overpartitions, which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition, and regular partition functions. We provide both generating function and bijective proofs. We also prove congruences for certain overpartition functions combinatorially.

Let \(G\) be a simple graph on \(n\) vertices. The Laplacian Estrada index of \(G\) is defined as \(LEE(G) = \sum_{i=1}^{n} e^{\mu_i}\), where \(\mu_1, \mu_2, \dots, \mu_n\) are the Laplacian eigenvalues of \(G\). In this paper, threshold graphs on \(n\) vertices and \(m\) edges having maximal and minimal Laplacian Estrada index are determined, respectively.

In this paper, formulas of the resistance distance for the arbitrary two-vertex resistance of \(G\), \(H = G_1 \boxdot G_2\) and \(G_1 \boxminus G_2\) in the electrical networks are obtained in a much simpler way. Furthermore, \(K_f(G_1 \boxdot G_2)\) and \(K_f(G_1 \boxminus G_2)\) can be expressed as a combination of \(K_f(G_1)\) and \(K_f(G_2)\).

Networks are important structures and appear in many different applications and settings. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Centrality parameters play an important role in the field of network analysis. Numerous studies have proposed and analyzed several centrality measures. These concepts measure the importance of a node’s position in a network. In this paper, vertex residual closeness \((VRC)\) and normalized vertex residual closeness \((NVRC)\) of some splitting networks modeled by splitting graphs are obtained.

Let \(T\) be an isosceles right triangle and let \(S_1, S_2, S_3, \dots\) be the homothetic copies of a square \(S\). In this paper, we consider the parallel covering and packing of \(T\) with the sequence \(\{S_n\}\) of squares.