Certain Classes Of Graphs Research Articles

A new spectral-based index for graphs and its application to polyaromatic compounds

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total [Formula: see text]-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

Partition energy of m-splitting graph and their generalized complements

Let G be graph with partition P={V1, V2, …, Vk} of the vertex set V. Recently E. Sampathkumar et al. [22] introduced the concept of k-partition energy of graph EPₖ (G) and studied partition energy of certain class of graphs. In this paper we compute partition energy of some m-splitting graphs Sm(G). The graphs we consider are Sm (Kn), Sm(Cn), Sm(Kn×2), Sm(Kn,n) and their generalized complements.

Rigidity on quantum symmetry for a certain class of graph C*-algebras

Quantum symmetry of graph C*-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group (C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ) [in short, *|E(Γ)|C(S1),Δ] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph C*-algebras remains *|E(Γ)|C(S1),Δ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with *|E(Γ)|C(S1),Δ. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

Normalized volumes of type-PQ adjacency polytopes for certain classes of graphs

Normalized volumes of type-PQ adjacency polytopes for certain classes of graphs

Minimum length word-representants of graph products

A graph G=(V,E) is said to be word-representable if a word w can be formed using the letters of the alphabet V such that for every pair of vertices x and y, x≠y, xy∈E if and only if x and y alternate in w. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles. It is known that Cartesian and rooted products preserve word-representability. Moreover, using occurrence-based functions, Broere constructed a uniform word representing the Cartesian product of G and Kn.In this paper, we study the minimum length of word-representants for Cartesian and rooted products using morphism and occurrence-based functions, respectively. Also, we solve an open problem posed by Broere in his master thesis. This problem asks to construct a word for the Cartesian product of two arbitrary word-representable graphs.

On Zagreb Energy of Certain Classes of Graphs

On Zagreb Energy of Certain Classes of Graphs

Using path coloring of graphs for communication in social networks

An L (p1, p2, p3, … , pm)- labeling of a graph G, has the vertices of G assigned with non-negative integers, such that the vertices at distance j should have at least pj as their label difference. If m = 3 and p1 = 3, p2 = 2, p3 = 1, it is called an L (3, 2, 1)-labeling which is widely studied in the literature. In this paper, we define an L (3, 2, 1)-path coloring of G as a labeling g : V (G) → Z+ such that between every pair of vertices there exists at least one path P where in the labeling restricted to this path is an L (3, 2, 1)-labeling. Among the labels assigned to any vertex of G under g, the maximum label is called the span of g. The L (3, 2, 1)-connection number of a graph G, denoted by k3c (G) is defined as the minimum value of span of g taken over all such labelings g. We call graphs with the special property that k3c (G) = |V (G) | as L (3, 2, 1)-path graceful. In this paper, we obtain k3c (G) of graphs that possess a Hamiltonian path and carry forward the discussion to certain classes of graphs which do not possess a Hamiltonian path, which is novel to this paper. Although different kinds of labeling are studied in the literature with different mathematical constraints imposed, the idea of showing the existence of a graph with a given number as its minimum labeling number has rarely been addressed. We show that given any positive integer, there always exists an L (3, 2, 1)-path graceful graph with the given integer as its k3c (G), thus addressing the inverse question. Finally exploiting the fact that there is no gap on the k3c (G) number line, we give an application of path colorings for secure communication on social networking sites. Efforts to deploy graph coloring in task scheduling, interference-free transmission, etc have been dealt by earlier researchers. In this paper, we deploy the L (3, 2, 1)-path coloring technique defined by us for secure communication in social networks, which has not been dealt with so far.

Some Bounds for the Regularity of the Edge Ideals and Their Powers in a Certain Class of Graphs

Some Bounds for the Regularity of the Edge Ideals and Their Powers in a Certain Class of Graphs

Bounds on the Clique and the Independence Number for Certain Classes of Graphs

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

Algebraic spectral analysis of the locating degree matrix of a graph

In this article, we present a new graph matrix based on locating degree, we call it the locating degree matrix. The spectrum and energy related to this new matrix are established for certain classes of graphs. Bounds of the new related energy for a special type of regular graph are obtained.

First-order logic with metric betweenness – the case of non-definability of some graph classes

The metric betweenness of a connected graph and arbitrary graphs is FO definable. Moreover, several interesting classes of graphs with strong distance properties are shown to be FO definable using metric betweenness. However, there are different classes that do not possess such an FO axiomatization. The main goal of this note is to show that certain classes of graphs do not allow an axiomatization in First-Order Logic with Betweenness (FOLB).

Average distance colouring of graphs

Abstract For a graph G with n vertices, average distance µ(G) is the ratio of sum of the lengths of the shortest paths between all pairs of vertices to the number of edges in a complete graph on n vertices. In this paper, we introduce average distance colouring and find the average distance colouring number of certain classes of graphs.

Minimum length word-representants of word-representable graphs

A graph G=(V,E) is said to be word-representable if a word w can be formed using the letters of the alphabet V such that for every pair of vertices x and y, xy∈E if and only if x and y alternate in w. Gaetz and Ji have recently introduced the notion of minimum length word-representants for word-representable graphs. They have also determined the minimum possible length of the word-representants for certain classes of graphs, such as trees and cycles.In this paper, we solved two of the three cases and partially solved the remaining case of an open problem presented in Gaetz and Ji (2020) by Gaetz and Ji. This problem asks to classify all 2-word-representable graphs based on the length of their minimum length word-representants. Furthermore, we found a relation between the maximum and minimum number of occurrences of letters in minimum length word-representants and the diameter of the graph. Finally, we gave an upper bound on the minimum possible length of a word-representant of a graph under certain word-representability preserving graph operations such as connecting two graphs by an edge and gluing two graphs in a vertex.

Bounds on the Twin-Width of Product Graphs

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

Rainbow independent sets in certain classes of graphs

AbstractRainbow matchings in graphs and in hypergraphs have been studied extensively, one motivation coming from questions on matchings in 3‐partite hypergraphs, including questions on transversals in Latin squares. Matchings in graphs are independent sets in line graphs, so a natural problem is to extend the study to rainbow independent sets in general graphs. We study problems of the following form: given a class of graphs, how many independent sets of size in a graph belonging to are needed to guarantee the existence of a rainbow set of size ? A particularly interesting case is the class of graphs having a given upper bound on their maximum degree.

Formula omitted]-distance secure sets in graphs

Let G=V,E be a graph and S be a non empty subset of V. For any s∈S, a vertex of Ns-S is an attacker of s and a vertex of Ns∩S is a defender of s. An attack on S is a collection of mutually disjoint sets of attackers of vertices in S. A defense of S is a collection of mutually disjoint sets of defenders of vertices in S. The set S is a secure set if for every attack on S, there is a defense of S such that the defending set has at least as many vertices as in the attacking set corresponding to every vertex of S. In this paper, k-distance secure sets are defined as a generalization secure sets. A characterization of k-distance secure sets is obtained. Further k-distance security numbers of certain classes of graphs are determined.

THE GEODETIC FAULT TOLERANT DOMINATION NUMBER OF A GRAPH

For a connected graph G = (V,E), a set F ⊆ V of vertices in G is called dominating set if every vertex not in F has at least one neighbor in F. A dominating set F ⊆ V is called fault tolerant dominating set if F − {v} is dominating set for every v ∈ F. A fault tolerant dominating set is said to be geodetic fault tolerant dominating set if I[F] = V . The minimum cardinality of a geodetic fault tolerant dominating set is called geodetic fault tolerant domination number and is denoted by γgft(G). The minimum geodetic fault tolerant dominating set is denoted by γgft-set. The geodetic fault tolerant domination number of certain classes of graphs are determined. Some general properties satisfied by this concept are studied. It is shown that for every positive integer 2 < a ≤ b there is a connected graph G such that γ(G) = a, γg(G) = b and γgft(G) = a + b − 2, where γ(G) and γg(G) are the domination number and geodetic domination number of G respectively.

The Upper Connected Outer Connected Monophonic Number of a Graph

For a connected graph of order at least two, a connected outer connected monophonic set of is called a minimal connected outer connected monophonic set if no proper subset of is a connected outer connected monophonic set of . The upper connected outer connected monophonic number of is the maximum cardinality of a minimal connected outer connected monophonic set of . We determine bounds for it and find the upper connected outer connected monophonic number of certain classes of graphs. It is shown that for any two integers with , there is a connected graph of order with and . Also, for any three integers and with , there is a connected graph with and and a minimal connected outer connected monophonic set of cardinality , where is the connected outer connected monophonic number of a graph.

Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.

Weak edge triangle free detour number of a graph

For any two vertices u and v in a connected graph G = ( V , E ), a u − v path P is called a u − v triangle free path if no three vertices of P induce a triangle. The triangle free detour distance D △ f ( u , v ) is the length of a longest u − v triangle free path in G . A u − v path of length D △ f ( u , v ) is called a u − v triangle free detour . A set S ⊆ V is called a weak edge triangle free detour set of G if every edge of G has both ends in S or it lies on a triangle free detour joining a pair of vertices of S . The weak edge triangle free detour number w d n △ f ( G ) of G is the minimum order of its weak edge triangle free detour sets and any weak edge triangle free detour set of order w d n △ f ( G ) is a weak edge triangle free detour basis of G . Certain properties of these concepts are studied. The weak edge triangle free detour numbers of certain classes of graphs are determined. Its relationship with the triangle free detour diameter is discussed and it is proved that for any three positive integers a , b and n of integers with 3 ≤ b ≤ n − a + 1 and a ≥ 4, there exists a connected graph G of order n with triangle free detour diameter D △ f = a and w d n △ f ( G )= b . It is also proved that for any three positive integers a , b and c with 3 ≤ a ≤ b and c ≥ b + 2, there exists a connected graph G such that R △ f = a , D △ f = b and w d n △ f ( G )= c .