A radio mean square labeling of a connected graph is motivated by the channel assignment problem for radio transmitters to avoid interference of signals sent by transmitters. It is an injective map h from the set of vertices of the graph G to the set of positive integers , such that for any two distinct vertices x, y, the inequality d(x, y) + ⌈ (h(x))² + (h(y))²/2 ⌉ ≥ dim(G) + 1 holds. For a particular radio mean square labeling h, the maximum number of h(v) taken over all vertices of G is called its spam, denoted by rmsn(h), and the minimum value of rmsn(h) taking over all radio mean square labeling h of G is called the radio mean square number of G, denoted by rmsn(G). In this study, we investigate the radio mean square numbers rmsn(P_n) and rmsn(C_n) for path and cycle, respectively. Then, we present an approximate algorithm to determine rmsn(G) for graph G. Finally, a new mathematical model to find the upper bound of rmsn(G) for graph G is introduced. A comparison between the proposed approximate algorithm and the proposed mathematical model is given. We also show that the computational results and their analysis prove that the proposed approximate algorithm overcomes the integer linear programming model (ILPM) according to the radio mean square number. On the other hand, the proposed ILPM outperforms the proposed approximate algorithm according to the running time.

In this paper, we used the permutation group together with the concept of cordiality in graph theory to introduce a new method of labeling. This construed permuted cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two paths and the union of any two cycles are permuted cordial graphs. In addition, we investigated the permuted cordiality for the union of any path with cycle.

A simple graph is called cordial if it admits 0‐1 labeling that satisfies certain conditions. The second order of lemniscate graph is a graph of two second order of circles that have one vertex in common. In this paper, we introduce some new results on cordial labeling, total cordial, and present necessary and sufficient conditions of cordial and total cordial for corona product of paths and second order of lemniscate graphs.

In this study, we propose a new kind of graph labeling which we call logic labeling and investigate the logically labeling of the corona between paths P_n and cycles C_n, namely, P_n⊙C_m. A graph is said to be logical labeling if it has a 0 − 1 labeling that satisfies certain properties. The corona G₁⊙G₂ of two graphs G₁ (with n₁ vertices and m₁ edges) and G₂ (with n₂ vertices and m₂ edges) is defined as the graph formed by taking one copy of G₁ and n₁ copies of G₂ and then connecting the i^th vertex of G₁ with an edge to every vertex in the i^th copy of G₂.

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path.

Radio antennas switch signals in the form of radio waves using different frequency bands of the electromagnetic spectrum. To avoid interruption, each radio station is assigned a unique number. The channel assignment problem refers to this application. A radio geometric mean labeling of a connected graph $G$ is an injective function h from the vertex set, $V\left(G\right)$ to the set of natural numbers $N$ such that for any two distinct vertices $x$ and $y$ of $G$, $\sqrt{h\left(x\right)·h\left(y\right)}\ge diam+1-d\left(x,y\right)$. The radio geometric mean number of $h$, $rgmn\left(h\right)$, is the maximum number assigned to any vertex of $G$. The radio geometric mean number of G, $rgmn\left(G\right)$ is the minimum value of $rgmn\left(h\right)$, taken over all radio geometric mean labeling $h$ of $G$. In this paper, we present two theorems for calculating the exact radio geometric mean number of paths and cycles. We also present a novel algorithm for determining the upper bound for the radio geometric mean number of a given graph. We verify that the upper bounds obtained from this algorithm coincide with the exact value of the radio geometric mean number for paths, cycles, stars, and bi-stars.