Radio Mean Labeling Algorithm, Its Complexity and Existence Results

Abstract

Radio mean labeling of a connected graph G is an assignment of distinct positive integers to the vertices of G satisfying a mathematical constraint called radio mean condition. The maximum label assigned to any vertex of G is called the $span$ of the radio mean labeling. The minimum $span$ of all feasible radio mean labelings of G is the radio mean number of G, denoted by $rmn\left(G\right)$. In our previous study, we proved that if G has order n, then $rmn\left(G\right)\in [n,rmn\left(P_n\right)]$ where $P_n$ is a path of order n. All graphs of diameters 1, 2 and 3 have the radio mean number equal to order n. However, they are not the only graphs on n vertices with radio mean number n. Graphs isomorphic to path $P_n$ are the graphs having the maximum diameter among the set of all graphs of order n and they possess the maximum feasible radio mean number. In this paper, we show that, for any integer in the range of achievable radio mean numbers, there always exists a graph of order n with the given integer as its radio mean number. This is approached by introducing a special type of tree whose construction is detailed in the article. The task of assigning radio mean labels to a graph can be considered as an optimization problem. This paper critiques the limitations of existing Integer Linear Programming (ILP) models for assigning radio mean labeling to graphs and proposes a new ILP model. The existing ILP model does not guarantee that the vertex labels are distinct, positive and satisfy the radio mean condition, prompting the need for an improved approach. We propose a new ILP model which involves $n^2$ constraints is the input graph’s order is n. We obtain a radio mean labeling of cycle of order 10 using the new ILP. In our previous study, we showed that, for any graph G, we can extend the radio mean labelings of its diametral paths to the vertex set of G and obtain radio mean labelings of G. This insight forms the basis for an algorithm presented in this paper to obtain radio mean labels for a given graph G with n vertices and diameter d. The correctness and complexity of this algorithm are analyzed in detail. Radio mean labelings have been proposed for cryptographic key generation in previous works, and the algorithm presented in this paper is general enough to support similar applications across various graph structures.

1. Introduction

For graphs and their associated theoretical concepts, we refer the reader to [1,2]. For graph labeling and its associated theoretical concepts, we refer the reader to [3,4]. graph labeling refers to assigning labels, which are elements of a particular set, to the elements of a graph G. Typically, this involves labeling the vertices, edges or both of G based on certain mathematical constraints. When only the vertices of G are assigned labels, G is called a vertex-labeled graph. When only the edges of G are labeled, G is called edge-labeled graph. Specifically, for a graph G, a vertex labeling of G is a function $l:V\left(G\right)⟶S$ and an edge labeling of G is a function $l^{\prime }:E\left(G\right)⟶S^{\prime }$, for some sets S and $S^{\prime }$. The combinatorial interest in the problem of graph labeling often arises from the mathematical constraints that are inherent in its definition.

The radio channel assignment problem is one of the most significant applications of vertex labeling of graphs with non-negative integers. The problem is to find a way to assign radio channels to transmitters, so neighboring channels have significant frequency differences, where channels far from one another might have minor frequency differences. This problem of radio channel assignment can be easily converted into a graph labeling problem by considering the radio network as a graph. In this case, the vertices correspond to transmitter locations, and two vertices are made adjacent if the locations of the radio stations corresponding to these vertices are close. The maximum interference occurs among transmitters represented by adjacent vertices. Assignment of frequencies to transmitters means the assignment of labels to vertices. The problem is to label (with colors or integers) the vertices of the graph so that the labels of adjacent vertices differ by a threshold, while the constrain does not hold for non-adjacent vertices. This concept has given rise to numerous distance-constrained graph coloring and labeling problems with varying constraints, whose primary goal is to minimize the maximum label assigned.

An early type of coloring inspired by the above concept is $L(h,k)$-coloring where $h,k$ are non-negative integers [5].

Definition 1

([5]). For a graph G, a $L(h,k)$-coloring is a function λ from $V\left(G\right)$ to a set of colors such that

1. 

if $d(u,v)=1$, $\left|\lambda \right(u)-\lambda (v\left)\right|\ge h$; 

2. 

if $d(u,v)=2$, $\left|\lambda \right(u)-\lambda (v\left)\right|\ge k$.

∀$u,v\in V\left(G\right)$ and some $h,k\in {\mathbb{Z}}^{+}$. The $span$ of λ is ${max}_{u,v\in V\left(G\right)}|\lambda \left(u\right)-\lambda \left(v\right)|$. The minimum $span$ of all $L(h,k)$- colorings of G is called the $L-span$ of G.

In 2001, Chartrand et al. [5,6] defined $k-$ radio coloring of graphs as follows:

Definition 2

([5,6]). For a connected graph G with diameter d and $k\in {\mathbb{Z}}^{+},1\le k\le d$, a radio $k-$ coloring (labeling) is a one-to-one function ϕ from $V\left(G\right)$ to ${\mathbb{Z}}^{+}$, where

$$d(u,v)+\left|\varphi \right(u)-\varphi (v\left)\right|\ge 1+k,\forall u,v\in V\left(G\right).$$

(1)

The maximum color (integer) assigned to any vertex of G by $\lambda$ is denoted by $r{c}_k\left(\lambda \right)$. The minimum $r{c}_k\left(\lambda \right)$ taken over all radio $k-$ colorings of G is called the $k-$ radio number of G.

The radio 1-coloring is a proper vertex coloring of G [5] and radio 2-coloring is an $L(2,1)$ coloring of G. The radio $d-$ coloring is called a radio labeling of G, defined as follows:

Definition 3

([6]). For a connected graph G with diameter d, a radio labeling is a one-to-one function ϕ from $V\left(G\right)$ to ${\mathbb{Z}}^{+}$, where

$$d(u,v)+\left|\varphi \right(u)-\varphi (v\left)\right|\ge 1+d,\forall u,v\in V\left(G\right).$$

(2)

The inequality (2) is called the radio condition.

The radio number of $\varphi$, $rn\left(\varphi \right)$, is the maximum number assigned to any vertex of G. The radio number of G, $rn\left(G\right)$, is the minimum value of $rn\left(\varphi \right)$ taken over all radio labelings $\varphi$ of G. Graphs for which the largest label used is the same as the order of the graph are called radio graceful graphs. The combinatorial problem of Radio mean labeling of a graph was proposed by R. Ponraj, S. Sathish Narayanan and R. Kala in [7].

Definition 4.

For a connected graph G with diameter d, a radio mean labeling is a one-to-one function γ from $V\left(G\right)$ to ${\mathbb{Z}}^{+}$, where for every pair of vertices $u,v$

$$d(u,v)+\lceil {\displaystyle \frac{\gamma \left(u\right)+\gamma \left(v\right)}{2}}\rceil \ge 1+d.$$

(3)

The inequality (3) is called the radio mean condition. The $span$ of $\gamma$ is the maximum integer assigned to any $v\in V\left(G\right)$ by $\gamma$. Furthermore, the radio mean number of G, denoted by $rmn\left(G\right)$, is the smallest $span$ of all feasible radio mean labelings of G. It is obvious by the definition that $rmn\left(G\right)\ge \left|V\right(G\left)\right|$. If $rmn\left(G\right)=\left|V\right(G\left)\right|$, then G is called a radio mean graceful graph [8]. See Figure 1 for some examples.

The radio mean number of subdivision of complete bipartite, corona complete graph with path and one point union of cycle $C_6$ were obtained in [9]. The radio mean number of corona product of wheel with path, the graph obtained from the wheels $W_m$ and $W_n$ by joining the rim vertices of the two wheels with an edge and the graph obtained from a wheel by subdividing each spoke by a vertex were obtained in [10]. The article [11] gave the radio mean number of the subdivisions of the graphs, namely star, wheel, join of $K_2$ and m copies of $K_1$, bistar and complete bipartite graph, corona product of complete bipartite graph with path, and friendship graph. In [12], the radio mean number of jelly fish graph and its subdivision, book with pentagonal pages, graphs obtained by taking m disjoint copies of $S_{1,n}$ and joining a new vertex to the centers of the m copies of $S_{1,n}$ were studied. K. Palani, S. S. Sabarina Subi and V. Maheswari [13] proved that the path union of graphs like star and its subdivision, globe and fan are radio mean graceful. They also obtained the radio mean number double fan graph. In [14], radio mean labeling of subdivisions of various graphs were discussed. The article [15] dealt with the radio mean numbers of splitting graph of graphs like path, star and Y-tree. In [16], K. Sunitha, C. David Raj and A. Subramanian studied the radio mean labelings of triangular ladder graph, corona product of path with $\overline{K_2}$, corona product of complete graph with $\overline{K_2}$ and corona product of wheel with $\overline{K_2}$. The radio mean labelings of degree splitting of certain graphs are studied in [17,18]. Upper bounds on the radio mean number of honeycomb and honeycomb torus networks were derived in [19]. In [20], M.S. Aasi, M. Asif, T. Iqbal and M. Ibrahim determined the radio mean number of lexicographic product of $P_p$ with $P_q$, where $p=2,3$. A Python program was also developed to compute the same. The authors of [21] proposed an integer linear programming model for the problem of radio mean labeling of graphs. They also put forward an algorithm to find an upper bound on the radio mean number of any graph. The radio mean numbers of path and cycle were also determined in [21]. Unfortunately, the authors of [21] made mistakes in this regard. Refs. [22,23] identified these inaccuracies and provided upper bounds on the radio mean number for cycles and paths. All the above-mentioned articles mostly focus on the combinatorial aspect of radio mean labeling, but articles [24,25,26] explored another side of it. They talked about different roles of graphs in cryptosystems: as a mode of data presentation and as a key generator.

1.1. Motivation of This Study

Finding a vertex labeling satisfying (3) for a graph is a challenging task, and so is labeling the graph with minimum $span$. Article [27] discussed the maximum and minimum achievable radio mean number of graphs of order n. In [27], it is proved that, if G is a connected graph on n vertices,

$$n\le rmn\left(G\right)\le rmn\left(P_n\right)$$

where $P_n$ is path on n vertices. An open question here is

Question 1.

For any given $n\in {\mathbb{Z}}^{+}$ and for every integer $r\in [n,rmn\left(P_n\right)]$, does there exist some graph G of order n such that $rmn\left(G\right)=r$?

In this article, we will provide answers to the above question.

Radio mean labeling of graphs typically involves defining a function on the vertex set that adheres to the radio mean condition. However, for graphs with a large order or diameter, this can be quite difficult. The real challenge in finding a radio mean labeling of a graph, then, becomes finding one with the smallest possible $span$. The next question that we attempt to solve in this article is as follows:  

Question 2.

What are the alternative methods for obtaining a radio mean labeling of a graph, instead of the traditional approach of defining a function on the vertex set, which can be time-consuming?

Radio mean labeling of a graph can be framed as an optimization problem involving positive integer variables subject to radio mean constraints. This article investigates the limitations of existing Integer Linear Programming (ILP) models, particularly their inability to ensure that the vertex labels are distinct, positive and satisfy the radio mean condition. We propose a new ILP model that involves $n^2$ constraints, where n is the order of the input graph. For any graph G, the radio mean labeling of its diametral paths can be extended to the entire vertex set to produce a valid labeling for G. This insight forms the foundation of the algorithm presented in this paper, which computes radio mean labelings for a graph G with n vertices and diameter d.

1.2. Organization of the Paper

The subsequent sections of this article are organized as follows. In Section 2, we show that every integer in the range of achievable radio mean numbers of connected graphs of order n is the radio mean number of some graph of order n. In Section 3, we propose two alternative methods for obtaining a radio mean labeling of any graph. We show in Section 3.1 that the problem of radio mean labeling of graphs can be formulated as an Integer Linear Program (ILP). Furthermore, in Section 3.2, we provide an algorithm for finding a radio mean labeling and study its complexity.

2. There Is No Gap in the Radio Mean Number Line

The $rmn\left(G\right)=\left|V\right(G\left)\right|$, for any graph G with $1\le diam\left(G\right)\le 3$. Hence, all graphs of diameters 1, 2 and 3 are radio mean graceful. However, these are not the only graphs on n vertices with radio mean number n. Characterization of such graphs is an open problem. Path $P_n$ on n vertices is the graph having maximum diameter among the set of all graphs of order n and they possess the maximum feasible radio mean number. In this section, we attempt to answer Question (1) using a particular type of tree whose construction is described below.

Let ${\mathbb{T}}_{n,d}$ be a tree with vertex set $V\left({\mathbb{T}}_{n,d}\right)=\{v_i:1\le i\le n\}$ with $n\ge 3$ and diameter $d>1$. Let $P:v_1{v}_2\dots v_{d+1}$, $2\le d\le n-1$ be a central or diametral path of ${\mathbb{T}}_{n,d}$. Note that the central or diametral path denotes the longest path contained within the graph. It is the path whose length matches the diameter of the graph. The vertices in $V\left({\mathbb{T}}_{n,d}\right)\setminus V\left(P\right)$ are made adjacent to $v_2$. Then, ${\mathbb{T}}_{n,d}$ is a Caterpillar tree as each $v\in V\left({\mathbb{T}}_{n,d}\right)\setminus V\left(P\right)$ is within distance 1 of P. See Figure 2 for an illustration.

Note that ${\mathbb{T}}_{n,2}$ is isomorphic to the star $S_n$ and ${\mathbb{T}}_{n,n-1}$ is isomorphic to the path $P_n$.

Let us now define an radio mean label for ${\mathbb{T}}_{n,d}$. If L denotes a function that assigns labels to vertices of ${\mathbb{T}}_{n,d}$, then following are the conditions the vertex labels must satisfy:

(i)

$\lceil {\displaystyle \frac{L\left(v_i\right)+L\left(v_j\right)}{2}}\rceil \ge d+1-d(v_i,v_j),\forall 1\le i,j\le d+1$

(ii)

$\lceil {\displaystyle \frac{L\left(v_i\right)+L\left(v_j\right)}{2}}\rceil \ge d-1,\forall d+2\le i<j\le n$

(iii)

$\lceil {\displaystyle \frac{L\left(v_i\right)+L\left(v_j\right)}{2}}\rceil \ge d-d(v_i,v_2),\forall 1\le i\le d+1,d+2\le j\le n$

As graphs of diameter 2 and 3 are radio mean graceful, above labelling for ${\mathbb{T}}_{n,d},d\in \{2,3\}$ is a graceful radio mean labeling and so $rmn\left({\mathbb{T}}_{n,d}\right)=n$.

Results on radio mean labeling of paths have been presented in [22] and we apply them here. Suppose $d\ge 4$. First, the diametral path P of ${\mathbb{T}}_{n,d}$ is radio mean labeled with minimum span. Based on the results in [22], let l be a radio mean labeling of P such that $span\left(l\right)=rmn\left(P\right)$. Let S be the set of all positive integers which are less than or equal to $rmn\left(P\right)$ but do not belong to the range set ${\mathcal{R}}_l$ of l. i.e., $S=\{1,2,\dots ,rmn\left(P\right)\}\setminus {\mathcal{R}}_l$.

If $d=4$, i.e., the order of P is 5, then P has a graceful radio mean labeling and $rmn\left(P\right)=5$. Then, S is empty.

If $d=5$, i.e., the order of P is 6, then $rmn\left(P\right)=7$. We can easily verify that if r is any radio mean labeling of P with $r\left(v\right)=1$ for some $v\in V\left(P\right)$, then $span\left(r\right)>7$. Also, if l is any radio mean labeling of P with minimum $span$, then ${\mathcal{R}}_l=\{2,3,\dots ,7\}$. Also, note that, in such a labeling l, if  $v_i$ and $v_j$ of P receive labels 2 and 3, then $d(v_i,v_j)$ must be 5 to satisfy the radio mean condition. Here, $S=\left\{1\right\}$. However, 1 cannot be assigned to any of the remaining unlabeled vertices of ${\mathbb{T}}_{n,5}$ satisfying the radio mean condition.

In a similar way, when $d\ge 6$, we can see that if l is a radio mean labeling of P of minimum $span$, then S is non empty but none of the elements of S can be assigned to any of the vertices of ${\mathbb{T}}_{n,d}\setminus V\left(P\right)$ satisfying the radio mean condition.

So, we define $L:V\left({\mathbb{T}}_{n,d}\right)⟶{\mathbb{Z}}^{+}$ by

$$L\left(v_j\right)=\left\{\begin{array}{c}l\left(v_j\right):1\le j\le d+1,\hfill \\ r+j-d-1:d+2\le j\le n,r=rmn\left(P\right).\hfill \end{array}\right.$$

Let $\{y_1,y_2,\dots ,y_n\}$ be an ordering of vertices of ${\mathbb{T}}_{n,d}$ such that $L\left(y_i\right)<L\left(y_j\right)$ whenever $i<j$. By the definition of L, it is clear that $y_i\in V\left(P\right)$ when $1\le i\le d+1$.

Consider the pair of vertices $(y_i,y_j)$ where $1\le i<j\le d+1$.

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{L\left(y_i\right)+L\left(y_j\right)}{2}}\rceil +d(y_i,y_j)& =\lceil {\displaystyle \frac{l\left(y_i\right)+l\left(y_j\right)}{2}}\rceil +d(y_i,y_j)\hfill \\ \hfill & \ge d+1.\hfill \end{array}$$

Consider the pair of vertices $(y_i,y_j)$ where $d+2\le i<j\le n$.

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{L\left(y_i\right)+L\left(y_j\right)}{2}}\rceil +d(y_i,y_j)& =\lceil r+{\displaystyle \frac{i+j}{2}}-d-1\rceil +2\hfill \\ \hfill & >d+1.\hfill \end{array}$$

Consider the pair of vertices $(y_i,y_j)$ where $1\le i\le d+1$ and $d+2\le j\le n$.

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{L\left(y_i\right)+L\left(y_j\right)}{2}}\rceil +d(y_i,y_j)& \ge \lceil {\displaystyle \frac{L\left(y_1\right)+r+j-d-1}{2}}\rceil +d(y_1,y_j)\hfill \\ \hfill & \ge \lceil {\displaystyle \frac{l\left(y_1\right)+l\left(y_{d+1}\right)}{2}}\rceil +\lceil {\displaystyle \frac{j-d-1}{2}}\rceil -1+2\hfill \\ \hfill & =\lceil {\displaystyle \frac{l\left(y_1\right)+l\left(y_{d+1}\right)}{2}}\rceil +\lceil {\displaystyle \frac{j-d-1}{2}}\rceil +1\hfill \\ \hfill & \ge d+2+\lceil {\displaystyle \frac{j-d-1}{2}}\rceil \hfill \\ \hfill & >d+1,\mathrm{as}j\ge d+2.\hfill \end{array}$$

Thus L is a radio mean labeling with $span\left(L\right)=rmn\left(P\right)+n-d-1$. Hence, $rmn\left({\mathbb{T}}_{n,d}\right)\le rmn\left(P\right)+n-d-1$. When $n\le 3$, ${\mathbb{T}}_{n,d}$ is isomorphic to $P_n$ and then $rmn\left({\mathbb{T}}_{n,d}\right)=rmn\left(P_n\right)=n$. For $n\ge 4$, the minimum feasible diameter of ${\mathbb{T}}_{n,d}$ is 2 and then $rmn\left({\mathbb{T}}_{n,d}\right)$ is n. The maximum diameter ${\mathbb{T}}_{n,d}$ can possess is $n-1$ and then ${\mathbb{T}}_{n,d}$ is isomorphic to a path of order n and hence $rmn\left({\mathbb{T}}_{n,d}\right)=rmn\left(P_n\right)$. By varying the value of d from 2 to $n-1$, it is seen that every integer within the range of achievable radio mean number is the radio mean number of some ${\mathbb{T}}_{n,d}\in \mathcal{T}$ where $\mathcal{T}$ is the collection of all Caterpillar trees ${\mathbb{T}}_{n,d}$ which are constructed as mentioned in the beginning of this section. To summarize,

Theorem 1.

Let $\mathcal{T}$ be the collection of all Caterpillar trees ${\mathbb{T}}_{n,d}$.

1. 

For ${\mathbb{T}}_{n,d}\in \mathcal{T}$, $n\le rmn\left({\mathbb{T}}_{n,d}\right)\le rmn\left(P_n\right)$.  

2. 

Every integer within the range of achievable radio mean number is the radio mean number of some ${\mathbb{T}}_{n,d}\in \mathcal{T}$.

Deriving a precise expression for the radio mean number of trees within the set $\mathcal{T}$ falls beyond the scope of this study, and the upper bound obtained earlier is sharp for most of the values of n and d.

Table 1. Radio mean number of ${\mathbb{T}}_{10,d}$ where $2\le d\le 9$.

Diameter, d	$\mathit{rmn}\left({\mathbb{T}}_{10,\mathit{d}}\right)$
2	10
3	10
4	11
5	11
6	12
7	13
8	14
9	14

provides the radio mean number of all Caterpillar trees ${\mathbb{T}}_{10,d}$ where $2\le d\le 9$. We conclude this section with the following Conjecture 1:

Conjecture 1.

For any given $n\in {\mathbb{Z}}^{+}$, every integer between n and $rmn\left(P_n\right)$is the radio mean number of some graph of order n, where $P_n$ denotes a path graph on n vertices.

3. Techniques for Generating Radio Mean Labeling

Two alternative ways of obtaining a radio mean labeling of a graph are discussed in this section. The radio mean labeling of a graph can be viewed as an optimization problem with positive integer variables subject to radio mean constraints, as discussed in Section 3.1. Taking advantage of the fact discussed in [27] that a radio mean labeling of a graph’s diametral path can be extended to the entire vertex set to produce a radio labeling of the graph, we present a polynomial-time algorithm in Section 3.2 for radio mean labeling of graphs. We designed an algorithm in Section 3.2 for radio mean labeling of any graph.

3.1. Integer Linear Programming Model for Radio Mean Labeling

In the article [21], the authors attempted to formulate the problem of generating radio mean labeling as an integer linear programming (ILP) model; however, the model presented is flawed, as explained below.

For a connected graph G with vertex set $V=\{v_i:1\le i\le n\}$ and diameter d and for $1\le i\le n$, let $x_i$ be the radio mean label assigned to vertex $v_i$ of G.

ILP 1

([21]). Minimize $L=x_1+x_2+\dots +x_n$ subject to the $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$ constraints

$$2n\lceil {\displaystyle \frac{x_i+x_j}{2}}\rceil \ge d+1-d(v_i,v_j)$$

for $1\le i\le n-1,2\le j\le n,i<j$ where $x_1,x_2,\dots ,x_n\in \{0,1\}$.

Note that $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)={\displaystyle \frac{n(n-1)}{2}}$. The model ILP 1 does not guarantee that the labels are distinct and are positive integers. Therefore, we propose a new ILP model for radio mean labeling of graphs.

The problem of finding a radio mean labeling G of diameter d is to find a function $\gamma$ such that $\gamma \left(v_i\right)=x_i,v_i\in V\left(G\right),x_i\in {\mathbb{Z}}^{+},1\le i\le n$ satisfying the radio mean condition and $\gamma \left(v_i\right)\ne \gamma \left(v_j\right)$ whenever $i\ne j$. The constraints are

$$\lceil {\displaystyle \frac{x_i+x_j}{2}}\rceil \ge d+1-d(v_i,v_j),x_i,x_j>0,x_i\ne x_j$$

where $1\le i\le n-1,2\le j\le n,i<j.$

Let $L=x_1+x_2+\dots +x_n$.

To ensure that the labels are positive integers:

For $1\le i<j\le n$, $x_i\ge 1.$

To ensure the uniqueness of labels:

For $1\le i<j\le n$, $x_i\ne x_j$

$⟹|x_i-x_j|\ge 1$

$⟹-(x_i-x_j)\ge 1$ or $x_i-x_j\ge 1$

$⟹-x_i+x_j\le -1.$

To ensure that the labels satisfy the radio mean condition:

$$\lceil {\displaystyle \frac{x_i+x_j}{2}}\rceil \ge d+1-d(v_i,v_j)$$

⟹${\displaystyle \frac{x_i+x_j}{2}}\ge d+1-d(v_i,v_j)$ or ${\displaystyle \frac{x_i+x_j+1}{2}}\ge d+1-d(v_i,v_j)$

⟹$x_i+x_j\ge (2d+2-2d(v_i,v_j))$ or $x_i+x_j\ge (2d+2-2d(v_i,v_j))-1$

⟹$x_i+x_j\ge 2d+2-2d(v_i,v_j)$

⟹$-x_i-x_j\le -2d+2d(v_i,v_j)-2$

Based on the above derivation, we now summarize the mathematical assumptions underlying the proposed ILP model:

• Integrality and Positivity: Each label $x_i$ is a positive integer, i.e., $x_i\in {\mathbb{Z}}^{+}$ and $x_i\ge 1$ for $1\le i\le n$.
• Distinctness of Labels: All labels must be distinct, i.e., $x_i\ne x_j$ for $i\ne j$, which is enforced using linear inequalities.
• Feasibility of the Radio Mean Condition: The labels must satisfy the radio mean constraint: $\lceil {\displaystyle \frac{x_i+x_j}{2}}\rceil \ge d+1-d(v_i,v_j)$ for all $i<j$.

Putting these facts together, we propose a new ILP model as follows:

ILP 2.

Minimize $L=x_1+x_2+\dots +x_n$ subject to the constraints

1. 

$-x_i-x_j\le -2d+2d(v_i,v_j)-2,\mathrm{where}1\le i\le n-1,2\le j\le n,i<j$

2. 

$-x_i+x_j\le -1,\mathrm{where}1\le i\le n-1,2\le j\le n,i<j$

3. 

$x_i\ge 1,x_i\in {\mathbb{Z}}^{+}$ where $1\le i\le n$

Solving this, we will obtain a radio mean labeling, say $\Gamma$, of G with $span(\Gamma )=max\{x_i:1\le i\le n\}$ and $span(\Gamma )$ gives an upper bound of the radio mean number of graph G.

Manually solving ILP 2 is not a viable option for graphs of large order, n, as it involves a significant number of constraints, specifically $n^2$. In the following example, we use MATLAB (https://matlab.mathworks.com) to solve the ILP model and find a radio mean labeling for the cycle, $C_{10}$. See Appendix A.

For $C_{10}$, let $V\left(C_{10}\right)=\{v_1,v_2,\dots ,v_{10}\}$, the diameter, $d=5$, and the distance matrix, $\mathbb{D}=\left[\begin{array}{cccccccccc}0& 1& 2& 3& 4& 5& 4& 3& 2& 1\\ 1& 0& 1& 2& 3& 4& 5& 4& 3& 2\\ 2& 1& 0& 1& 2& 3& 4& 5& 4& 3\\ 3& 2& 1& 0& 1& 2& 3& 4& 5& 4\\ 4& 3& 2& 1& 0& 1& 2& 3& 4& 5\\ 5& 4& 3& 2& 1& 0& 1& 2& 3& 4\\ 4& 5& 4& 3& 2& 1& 0& 1& 2& 3\\ 3& 4& 5& 4& 3& 2& 1& 0& 1& 2\\ 2& 3& 4& 5& 4& 3& 2& 1& 0& 1\\ 1& 2& 3& 4& 5& 4& 3& 2& 1& 0\end{array}\right].$

The ILP model for the radio mean labeling of $C_{10}$ is given by ILP 3.

ILP 3.

Minimize $L=x_1+x_2+\dots +x_{10}$ subject to the constraints

1. 

$-x_i-x_j\le -12+2d(v_i,v_j), \mathit{where} 1\le i\le 9,2\le j\le 10,i<j.$

2. 

$-x_i+x_j\le -1, \mathit{where} 1\le i\le 9,2\le j\le 10,i<j.$

3. 

$x_i\ge 1,x_i\in {\mathbb{Z}}^{+}$ where $1\le i\le 10.$

Manually solving the above ILP that includes 100 constraints is challenging. Hence, we make use of the function $intlinprog$ available in MATLAB. The syntax for Mixed-integer linear programming (MILP) in MATLAB is

$$x=intlinprog(f,intcon,A,b,Aeq,beq,lb,ub).$$

For $C_{10}$, the input arguments are

• f: coefficient vector representing objective function = $\left[1111111111\right]$,
• $intcon$: a vector that indicates integer-constrained variables = $[1,2,3,4,5,6,7,8,9,10]$,
• A: a matrix representing the linear inequality constraints
     =$\left[\begin{array}{cccccccccc}-1& -1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ ⋮& & & ⋮& & & ⋮& & & ⋮\\ 0& 0& 0& 0& 0& 0& 0& -1& 0& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 1\end{array}\right],$
• b: column vector representing the right-hand side of the inequality constraints = $\left[\begin{array}{c}-10\\ -8\\ -6\\ -4\\ -2\\ -4\\ ⋮\\ -1\\ -1\\ -1\end{array}\right],$
• $Aeq$: a matrix representing the linear equality constraints = $\left[\right],$
• $beq$: a column vector representing the right-hand side of the equality constraints = $\left[\right],$
• $lb$: a column vector representing the lower bounds on the variables = $\left[1111111111\right],$
• $ub$: a column vector representing the upper bounds on the variables = $\left[\right].$
• The output obtained is as follows:
• $x=\left[\begin{array}{c}14\\ 13\\ 12\\ 11\\ 10\\ 9\\ 8\\ 7\\ 6\\ 4\end{array}\right].$

3.2. Radio Mean Labeling Algorithm and Its Complexity

Let G be a connected graph of order n and diameter d having vertex set $V\left(G\right)=\{v_1,v_2,v_3,\dots ,v_n\}$. Let $P:v_1{v}_2\dots v_{d+1}$ be a diametral path of G and $\mathbb{D}$ be the distance matrix of G. As demonstrated in [27], if l is a radio mean labeling of P, the extension of l to $V\left(G\right)$ gives a radio mean labeling of G. The radio mean labeling of a path graph has already been studied in [22], and the labeling discussed there can be utilized. In this section, we provide an algorithm to label the vertices of G in a way that satisfies the radio mean condition, which is essentially an extension of the labeling discussed in [22].

The outline of the proposed algorithm is given in Figure 3. It takes vertex set of G, vertex set of diametral path P, diameter d of G, distance matrix $\mathbb{D}$ as input and returns vertices of G together with its labels and the $span$ of this labeling. Initially, all vertices are assigned 0 as the label. Then, the diametral path of G is radio mean labeled. Later, the vertices not in the (labeled) diametral path are assigned labels in such a way that radio mean condition holds between every pair of distinct vertices of G.

Algorithm 1 uses the radio mean labeling of paths discussed in [22] to label the diametral path P. There may exist many other radio mean labelings of P with the same or lesser $span$. Suppose that S in step 47 is nonempty. In such a case, we must check for all possible ways of labeling vertices of $V\left(G\right)\setminus V\left(P\right)$ with elements of S and maximum number of elements of S must be used for labeling vertices. This ensures that radio mean labeling of G obtained is having lesser span. Also, in step 68, if U is nonempty, there is no specific order in assigning labels to the vertices in U. One may produce different radio mean labelings of G using this algorithm. The $span$ of each of these labelings is an upper bound of the radio mean number of G. The minimum among these $spans$ will be a better upper bound.

Algorithm 1 To find a radio mean labeling of a connected graph G.

Input: $d,V\left(G\right),V\left(P\right),\mathbb{D}$ Output: vertices together with labels, $span$ Begin:    1: $n\leftarrow \left|V\right(G\left)\right|$    2: for j← 1 to n do    3:       $R\left(v_j\right)\leftarrow 0$    4: end for    5: if $d+1\in \{2,3\}$then    6:       for $j\leftarrow 1$ to $d+1$ do    7:             $R\left(v_j\right)\leftarrow j$    8:       end for    9: end if  10: if $d+1\in \{4,5,6\}$ then  11:       $R\left(v_1\right)\leftarrow d-2$  12:       $R\left(v_2\right)\leftarrow 2d-2$  13:       $R\left(v_3\right)\leftarrow d-1$  14:       for $j\leftarrow 4$ to $d+1$ do  15:             $R\left(v_j\right)\leftarrow 2d+1-j$  16:       end for  17: end if  18: if $d+1\ge 7$ then  19:       $m\leftarrow 1$  20:       while $d+1\notin [4+{\displaystyle \frac{m(m+5)}{2}},6+{\displaystyle \frac{m(m+5)}{2}}+m]$ do  21:             $m\leftarrow m+1$  22:       end while  23:       $p_1\leftarrow 1$  24:       $R\left(v_{p_1}\right)\leftarrow d-m-2$  25:       for $j\leftarrow 2$ to $m+2$ do  26:             $p_j\leftarrow d+p_{j-1}-R\left(v_{p_{j-1}}\right)$  27:             $R\left(v_{p_j}\right)\leftarrow d+j-m-3$  28:       end for  29:       for $j\leftarrow p_{m+2}+1$ to $d+1$ do  30:             $R\left(v_j\right)\leftarrow 2d-j+1$  31:       end for  32:       $l\leftarrow m+2$  33:       while $l\ge 2$ do  34:             for $j\leftarrow p_{(l-1)}+1$ to $p_l-1$ do  35:                   $R\left(v_j\right)\leftarrow R\left(v_{p_l+1}\right)-j+p_l$  36:             end for  37:             $l\leftarrow l-1$  38:       end while  39: end if  40: $r\leftarrow max\{R\left(v_j\right):v_j\in V\left(P\right)\}$  41: if $\left|V\right(P\left)\right|==n$ then  42:       go to step 71  43: end if  44: $U\leftarrow \{u\in V(G):R(u)=0\}$  45: $V^{\prime }\leftarrow \{v\in V\left(G\right):R\left(v\right)\ne 0\}$  46: $S\leftarrow \{1,2,\dots ,r\}\setminus \{R\left(v_1\right),R\left(v_2\right),\dots ,R\left(v_{d+1}\right)\}$  47: if $S\ne \varnothing$ then  48:       for each $s\in S$ do  49:             for each $u\in U$ do  50:                   $temp\left(u\right)\leftarrow s$  51:                   for each $v\in V^{\prime }$ do  52:                         if $\lceil {\displaystyle \frac{temp\left(u\right)+R\left(v\right)}{2}}\rceil +d(u,v)\ge d+1$ then  53:                               $rml\_condn\leftarrow Y$  54:                         else  55:                               $rml\_condn\leftarrow N$  56:                               Break and Go to Step 50  57:                         end if  58:                   end for  59:                   if $rml\_condn\leftarrow Y$ then  60:                         $R\left(u\right)\leftarrow temp\left(u\right)$  61:                         $U\leftarrow U\setminus \left\{u\right\}$  62:                         $V^{\prime }\leftarrow V\bigcup \left\{u\right\}$  63:                   end if  64:             end for  65:       end for  66: end if  67: if $U\ne \varnothing$ then  68:       for each $u\in U$ do  69:             $r\leftarrow r+1$  70:             $R\left(u\right)\leftarrow r$  71:       end for  72: end if  73: $span\leftarrow r$  74: return $\left\{\right(v,R\left(v\right)):v\in V(G\left)\right\}$, $span$ End

3.2.1. The Correctness of Algorithm 1

It is clear that the algorithm assigns distinct positive integers as labels to the vertices of G. Let us now check whether every pair of vertices in G satisfies the radio mean condition. As mentioned earlier, radio mean labeling discussed in [22] is exploited to label vertices belonging to P and so the labels assigned to any pair of vertices $v_i,v_j\in V\left(P\right)$ by this algorithm satisfies radio mean condition. Let $S=\{1,2,\dots ,r\}\setminus \left\{R\right(v):v\in V(P\left)\right\}$ where $r=max\left\{R\right(v):v\in V(P\left)\right\}$. If S is non empty, the algorithm itself checks if the vertices of $V\left(G\right)\setminus V\left(P\right)$ can be labeled with elements of S in such a way that radio mean condition holds between every pair of distinct vertices of G. Finally, let $U^{\prime }=\{v\in V\left(G\right):R\left(v\right)>r\}$. Note that if $v\in U^{\prime }$, $R\left(v\right)=r+q$ for some integer q where $1\le q\le |U^{\prime }|$.

Consider any pair of vertices $v_i,v_j\in U^{\prime }$. Then,

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{R\left(v_i\right)+R\left(v_j\right)}{2}}\rceil +d(v_i,v_j)& =\lceil {\displaystyle \frac{r+q+r+q^{\prime }}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & \ge r+\lceil {\displaystyle \frac{q+q^{\prime }}{2}}\rceil -1+d(v_i,v_j)\hfill \\ \hfill & \ge d+1,\mathrm{as}r\ge d.\hfill \end{array}$$

Consider a pair of vertices $(v_i,v_j)$ such that $v_i\in V\left(P\right)$ and $v_j\in U^{\prime }$.

If $2\le d+1\le 3$, then $r=R\left(v_{d+1}\right)$ and

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{R\left(v_i\right)+R\left(v_j\right)}{2}}\rceil +d(v_i,v_j)& =\lceil {\displaystyle \frac{R\left(v_i\right)+r+q}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & =\lceil {\displaystyle \frac{R\left(v_i\right)+R\left(v_{d+1}\right)+q}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & =\lceil {\displaystyle \frac{i+d+1+q}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & \ge d+1.\hfill \end{array}$$

If $d+1\in [4+{\displaystyle \frac{m(m+5)}{2}},6+{\displaystyle \frac{m(m+5)}{2}}+m],m=0,1,2,\dots$, then $r=R\left(v_2\right)$ and

$$\begin{array}{cc}\hfill \lceil {\displaystyle \frac{R\left(v_i\right)+R\left(v_j\right)}{2}}\rceil +d(v_i,v_j)& =\lceil {\displaystyle \frac{R\left(v_i\right)+r+q}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & =\lceil {\displaystyle \frac{R\left(v_i\right)+R\left(v_2\right)+q}{2}}\rceil +d(v_i,v_j)\hfill \\ \hfill & \ge \lceil {\displaystyle \frac{3d-2m-4}{2}}\rceil +\lceil {\displaystyle \frac{q}{2}}\rceil -1+d(v_i,v_j)\hfill \\ \hfill & =\lceil {\displaystyle \frac{3d}{2}}-m-2\rceil +\lceil {\displaystyle \frac{q}{2}}\rceil -1+d(v_i,v_j)\hfill \\ \hfill & \ge d+1,\mathrm{as}d>>m.\hfill \end{array}$$

Thus, the radio mean condition is satisfied by every pair of vertices of G and this shows that Algorithm 1 provides a radio mean labeling of G.

3.2.2. On the Complexity of Algorithm 1

Let us now estimate the computational complexity of this algorithm.

• The first step of this algorithm is to loop through $V\left(G\right)$ and assign 0 as label to each of the n vertices. Hence, the computational complexity of steps 2–5 is $O\left(n\right)$.
• Next, the vertices of diametral path P of G are assigned labels. As we are interested in worst-case complexity, we count only the complexity when $d+1\ge 7$. The complexity of steps $19-40$ is $O\left(m\right)+O(m+1)+O(d+1)+O(m+2)\equiv O\left(d\right),\mathrm{as}m<<d$. In worst case, $d=n-1$ and then complexity is $O\left(n\right)$.
• Each of the steps 41–47 is of complexity $O\left(1\right)$.
• After labeling the diametral path, we check whether any of the integers of the set S can be assigned to vertices not belonging to P satisfying Radio mean condition. Complexity of steps 48–67 is $O(d-m-3)\times O(n-d-1)\times O(d+1)\equiv O\left(n{d}^2\right)$. Note that steps from 41 on wards will be considered for execution only if $d\le n-2$. To evaluate the worst-case complexity, we need to assume that $d=n-2$ and in such a case complexity of steps 48–67 together is of order $n^2$.
• A maximum of $n-d-1$ vertices can still be there with label 0 and they will be labeled next with positive integers. The complexity of steps 68–73 is $O(n-d-1)$. To analyze the worst case, we must consider the minimum value of d and that is 1. Hence, the complexity of steps 68–73 together is of order n.
• Each of the steps 74–75 is of complexity $O\left(1\right)$.

Combining all the above steps, we see that total computational complexity, $T\left(n\right)=O\left(n\right)+O\left(n\right)+5\times O\left(1\right)+O\left(n^2\right)+O\left(n\right)+2\times O\left(1\right)\equiv O\left(n^2\right)$.

3.2.3. A Demonstration of Algorithm 1

We shall now illustrate how this algorithm works.

Let us consider a cycle $C:v_1{v}_2\dots v_{19}{v}_{20}$. Then, C has diameter 10 and let $P:v_1{v}_2\dots v_{10}{v}_{11}$ be a diametral path of C. The algorithm will initialize the label as 0 for each vertex of C. Since diameter of C is greater than 6, steps from 6 to 18 of Algorithm 1 are skipped. As $d+1=11$ belongs to the interval $[11,15]$, we have $m=2$. The vertices belonging to path P will be then assigned labels as follows: $v_1\leftarrow 6,v_2\leftarrow 16,v_3\leftarrow 15,v_4\leftarrow 14,v_5\leftarrow 7,v_6\leftarrow 13,v_7\leftarrow 12,v_8\leftarrow 8,v_9\leftarrow 11,v_{10}\leftarrow 9,v_{11}\leftarrow 10$. Thus, $r=16$, $U=\{v_{12},v_{13},\dots ,v_{20}\}$, $V=\{v_1,v_2,\dots ,v_{11}\}$ and $S=\{1,2,3,4,5\}$. The algorithm will now verify whether any of the integers of S can be assigned to a vertex in U in such a way that radio mean condition holds between every pair of distinct vertices of C. Having labeled $v_1,v_2,\dots ,v_{11}$ in this way, it is seen that none of the remaining vertices can be labeled with 1 or 2 satisfying radio mean condition. However, one of the following is possible: $\left(i\right)$ $v_{15}\leftarrow 3$, $\left(ii\right)$ $v_{15}\leftarrow 4$, $\left(iii\right)$ $v_{14}\leftarrow 5$. Let us label $v_{15}$ with 3. So U is now the set $\{v_{12},v_{13},v_{14},v_{16},\dots ,v_{20}\}$. Since $v_{15}$ obtains the label 3, integers 4 and 5 cannot be used to label the vertices of U. These vertices of U will now be assigned integers $\{17,18,\dots ,24\}$. There are 8 vertices in U and so there are $8!$ different ways of labeling them with distinct integers of the set $\{17,18,\dots ,24\}$. One such labeling is $v_{12}\leftarrow 24$, $v_{13}\leftarrow 23$, $v_{14}\leftarrow 22$, $v_{15}\leftarrow 3$, $v_{16}\leftarrow 21$, $v_{17}\leftarrow 20$, $v_{18}\leftarrow 19$, $v_{19}\leftarrow 18$, $v_{20}\leftarrow 17$. Therefore, $span$ of this radio mean labeling of C is 24. Hence, the output of Algorithm 1 is as follows:

$$\begin{array}{c}\{(v_1,6),(v_2,16),(v_3,15),(v_4,14),(v_5,7),\hfill \\ (v_6,13),(v_7,12),(v_8,8),(v_9,11),(v_{10},9),\hfill \\ (v_{11},10),(v_{12},24),(v_{13},23),(v_{14},22),(v_{15},3),\hfill \\ (v_{16},21),(v_{17},20),(v_{18},19),(v_{19},18),(v_{20},17)\},24.\hfill \end{array}$$

See Figure 4 for the intermediate stages of radio mean labeling $C_{20}$ using Algorithm 1. Note that the output of Algorithm 1 is not unique.

3.2.4. Comparing Algorithms

Let us now compare the algorithm to obtain an upper bound on the radio mean number proposed in [21] with Algorithm 1. The algorithm of [21] begins by labeling the first vertex of the input graph with its diameter d. It then assigns labels to the remaining vertices, ensuring that each vertex receives the next minimum possible integer as label while satisfying the radio mean constraint. For a graph of order n, the upper bound of radio mean number returned by this algorithm is $n+d$. However, Algorithm 1 in Section 3.2 produces a labeling with span as follows:

• $span=n:\mathrm{if}d=1,2,3$;
• $span\le n+1:\mathrm{if}d=4,5$;
• $span\le n+d-k-3:\mathrm{if}d=6,7,8,9,\dots ,$$d+1\in [4+S_k,6+S_k+k],S_k={\displaystyle \frac{k(k+5)}{2}} \mathrm{and} k=1,2,3,\dots$.

Clearly, Algorithm 1 gives improved upper bounds on the radio mean number of graphs.

4. Concluding Remarks

Since all graphs of diameter $1,2,3$ have a radio mean number equal to their order, it is obvious that, given a positive integer, there is always a graph with that integer as radio mean number. When we showed that there is no gap in the radio mean number line, it also proved the existence of graphs with diameter greater than 3 with a given integer as radio mean number.

The radio mean condition of a graph is solely dependent on its diameter, which makes finding suitable labeling for graphs with high order and diameter challenging. It becomes even more complex when dealing with graph classes that have order-dependent diameters. Designing an algorithm or ILP model that can radio mean label a specific class of graph with minimum span may be less arduous. Nevertheless, creating the same that can efficiently radio mean label any arbitrary graph with minimum span remains challenging.

In their publication [25], the authors proposed a novel method for generating cryptographic keys using radio mean labeled graphs. The cryptoalgorithm in scope of their study was Triple DES algorithm. The chosen graph for this key generation process was the Caterpillar tree, which had been previously discussed in Section 2. Algorithm 1 outlined in Section 3.2 is designed with the flexibility to generalize key generation techniques, enabling its adaptation to various graph structures beyond the Caterpillar tree. This method represents a significant advancement in cryptographic key generation methodologies, a topic that is not within the focus of the current study but certainly merits further exploration.

While an ILP model and a polynomial-time algorithm have been proposed to obtain radio mean labeling, determining the exact radio mean number of a given graph remains an open problem. Currently, the only way to find the radio mean number of a graph is to enumerate all feasible radio mean labelings and identify the one with the minimum $span$. This observation serves as a motivation to explore the integration of radio mean labeling with cryptographic algorithms aiming for more secure communication. This integration could possibly add additional layers of protection on the keys and encrypted message.

Despite our study’s reliance on a tailored custom algorithm, the optimization field offers a diverse range of sophisticated techniques worthy of examination for their effectiveness. Advanced optimization algorithms, such as hybrid heuristics and meta heuristics, discussed in [28,29,30,31,32], have demonstrated their efficacy in tackling complex decision problems across various domains by efficiently exploring solution spaces. These algorithms have broad applications beyond our study’s scope, encompassing areas such as online learning, scheduling, transportation, and data classification, showcasing their versatility and promise in addressing real-world challenges. Future research endeavors could involve comparative analyses to assess our proposed approach against advanced optimization algorithms, offering insights into their respective strengths and weaknesses.