Radio Labeling of Double Triangular Snake Graphs

Graph labelling problem has been broadly studied for a long period for its applications, especially in frequency assignment in (mobile) communication system, X -ray crystallography, circuit design, etc. Nowadays, surjective L 2,1 -labelling is a well-studied problem. Motivated from the L 2,1 -labelling problem and the importance of surjective L 2,1 -labelling problem, we consider surjective L 2,1 -labelling ( SL 21 -labelling) problems for paths and interval graphs. For any graph G = V , E , an SL 21 -labelling is a mapping φ : V ⟶ 1,2 , … , n so that, for every pair of nodes u and v , if d u , v = 1 , then φ u − φ v ≥ 2 ; and if d u , v = 2 , then φ u − φ v ≥ 1 , and every label 1,2 , … , n is used exactly once, where d u , v represents the distance between the nodes u and v , and n is the number of nodes of graph G . In the present article, it is proved that any path P n can be surjectively L 2,1 -labelled if n ≥ 4 , and it is also proved that any interval graph IG G having n nodes and degree Δ > 2 can be surjectively L 2,1 -labelled if n = 3 Δ − 1 . Also, we have designed two efficient algorithms for surjective L 2,1 -labelling of paths and interval graphs. The results regarding both paths and interval graphs are the first result for surjective L 2,1 -labelling.

An L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

[Formula: see text]-labeling problem ([Formula: see text]-[Formula: see text]) is an important topic in discrete mathematics due to its various applications, like in frequency assignment in mobile communication systems, signal processing, circuit design, etc. [Formula: see text]-[Formula: see text] is a special case of [Formula: see text]-[Formula: see text]. An [Formula: see text]-labeling ([Formula: see text]) of a graph [Formula: see text] is a mapping [Formula: see text] such that [Formula: see text] if and only if [Formula: see text], [Formula: see text] if [Formula: see text] or 3, where [Formula: see text] is the distance between the nodes [Formula: see text] and [Formula: see text]. In this work, we have determined the upper bound of [Formula: see text] for interval graph (IG) and obtained a tighter upper bound which is [Formula: see text]. Also, we proposed an efficient algorithm to label any IG by [Formula: see text] and also computed the time complexity of the proposed algorithm.

In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

L ( 2 , 1 ) -labeling is a graph coloring model inspired by a frequency assignment in telecommunication. It asks for such a labeling of vertices with nonnegative integers that adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. It is known that for any k ⩾ 4 it is NP-complete to determine if a graph has a L ( 2 , 1 ) -labeling with no label greater than k. In this paper we present a new bound on complexity of an algorithm for finding an optimal L ( 2 , 1 ) -labeling (i.e. an L ( 2 , 1 ) -labeling in which the largest label is the least possible). We improve the upper complexity bound of the algorithm from O ⁎ ( 3.5616 n ) to O ⁎ ( 3.2361 n ) . Moreover, we establish a lower complexity bound of the presented algorithm, which is Ω ⁎ ( 3.0739 n ) .

Coloring immersion-free graphs

Graph labeling is a well-known theme of graph theory that involves an assignment of integers to the domain elements such as vertices or edges, or both, subject to certain conditions. A bibliometric and descriptive quantitative approach is used in this study to conduct a bibliometric analysis on graph labeling by integrating mapping analysis with VOSviewer software. The data was obtained from a Google Scholar search using the keyword "graph labeling" that resulted in 980 articles published between 2018 and 2023, but only 375 of these articles were relevant to the subject. The results show that research on graph labeling changed from 2018 to 2023. To sum up, this work is the first to use VOSviewer for bibliometric analysis of a graph labeling study. It is hoped that this will make it a useful resource for future research on related issues.

A radio harmonic mean labeling of a connected graph G is a one to one map f from the vertex set V(G) to the set of natural numbers N such that for two distinct vertices u and v of G, d(u,v)+[ 2f(u)f(v)f(u)+f(v) ]≥1+diam(G). The radio harmonic mean number of f, rhmn(f) is the maximum number assigned to any vertex of G. The radio harmonic mean number of G, rhmn(G) is the minimum value of rhmn(f) taken over all radio harmonic mean labeling f of G. In this paper we have determined the radio harmonic mean number of some star related graphs.

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

The co-maximal subgroup graph Γ(G), of a finite group G, is a special kind of undirected graph, in which the set of vertices contains non-trivial proper subgroups of G and two distinct vertices A and U are adjacent iff AU = G. In this paper, we discuss the structures of co-maximal subgroup graphs of finite cyclic groups ℤpn, ℤp1np2m, ℤp1np2p3, ℤpn1pm2p3 and ℤp1np2mpl3 for n, m, l ∈ N, where p1, p2 and p3 are distinct prime numbers. Also, we establish the super graceful labeling of these graphs.

In this paper, we first establish a Multi-Relation Granular Computing model by a given graph, and point out that all the reducts of the constructed Multi-Relation Granular Computing model are exactly all the minimal vertex covers of the corresponding graph. Thus, the vertex cover problem in graph theory can be converted to the knowledge reduction problem in rough set theory. Based on the conversion, we then introduce methods for dealing with the knowledge reduction problem to solve the vertex cover problem. In particular, we introduce a kind of method called the heuristic reduction algorithm based on entropy.

Vertex guarding is one of many optimisation problems in graph theory with wide area of applications. It is proven to be NP-hard, therefore fast approximative solutions are significant. In the paper, at first, known algorithms are considered, and then a new algorithm working on planar graphs is introduced. The new algorithm is based on the dynamic approach and produces better and faster solutions. Its efficiency among other algorithms is demonstrated experimentally. In addition, ideas to additionally improve the algorithm are presented at the end.

Since its appearance, the number of spanning trees of a graph has been among the most important problems in graph theory. We aimed to get explicit formula counting this number in the corona product graph of two planar graphs. In this paper, we study the corona product of a planar graph with a linear chain and cycle chain, for which we calculate their number of spanning trees. Our research findings highlight the potential of combinatorial method, which allowed us to count this number for a large graph as corona product graph.

The Protein Data Bank currently contains about 600 data sets of three-dimensional protein coordinates determined by X-ray crystallography or NMR. There is considerable redundancy in the data base, as many protein pairs are identical or very similar in sequence. However, statistical analyses of protein sequence-structure relations require nonredundant data. We have developed two algorithms to extract from the data base representative sets of protein chains with maximum coverage and minimum redundancy. The first algorithm focuses on optimizing a particular property of the selected proteins and works by successive selection of proteins from an ordered list and exclusion of all neighbors of each selected protein. The other algorithm aims at maximizing the size of the selected set and works by successive thinning out of clusters of similar proteins. Both algorithms are generally applicable to other data bases in which criteria of similarity can be defined and relate to problems in graph theory. The largest nonredundant set extracted from the current release of the Protein Data Bank has 155 protein chains. In this set, no two proteins have sequence similarity higher than a certain cutoff (30% identical residues for aligned subsequences longer than 80 residues), yet all structurally unique protein families are represented. Periodically updated lists of representative data sets are available by electronic mail from the file server "netserv@embl-heidelberg.de." The selection may be useful in statistical approaches to protein folding as well as in the analysis and documentation of the known spectrum of three-dimensional protein structures.

Graph labeling was first introduced by Rosa in 1966. Labeling of graphs is an assignment of nonnegative integers to vertices, edges or both according to some specified conditions. Mean labeling of graphs was introduced by Somasundaram and Ponraj in 2003. Subsequently, labelings of graphs were done with geometric mean, harmonic mean etc. In this paper we introduce two concepts called ’Logarithmic mean labeling’, ’Identric mean labeling’ and acquire their mean labelings for some standard graphs.