Vertices Of Degree Research Articles

Radio labeling of biconvex split graphs

ABSTRACT A radio labeling of a graph G is a function f : V ( G ) → { 0 , 1 , … } such that for every pair of distinct vertices u , v ∈ V ( G ) , | f ( u ) − f ( v ) | ≥ 1 + diam ( G ) − d ( u , v ) , where diam(G) denotes the diameter of the graph and d(u, v) is the distance between the vertices u and v. The span of a radio labeling f of a graph G is the difference between the least and the largest labels assigned by f and is denoted by, span(f). The radio number of a graph G denoted by, rn(G), is the least positive integer s, such that there exists a radio labeling of G with span s. In this paper, we study the radio labeling of a special class of split graphs called biconvex split graphs of diameter three and we obtain both a lower bound and an upper bound for the radio number of biconvex split graphs of diameter three. Further, we determine the radio number of biconvex split graphs with three maximum degree vertices having disjoint independent neighbors.

Use Lexicographic Max Product of Picture Fuzzy Graph in Human Trafficking

Graph structures are an essential tool for solving combinatorial problems in computer science and computational intelligence. With an emphasis on signed graphs, picture-fuzzy graphs, and graphs with colored or labeled edges, this study explores the properties of picture-fuzzy graph topologies. Within these frameworks, it presents key ideas such as the lexicographic-max product, vertex degree, and total degree. The use of picture-fuzzy graphs' lexicographic-max product to tackle intricate problems like human trafficking is a key component of this study. The study illustrates how this strategy can improve decision-making processes in such crucial areas by utilizing the special qualities of picture-fuzzy graphs. The study is supported by informative numerical examples that show how useful these ideas are in real-world situations. In addition, the study offers a thorough algorithmic foundation for applying the lexicographic-max product in practical situations, especially those involving human trafficking. The goal of this framework is to provide a workable approach for applying picture-fuzzy graph structures to enhance decision-making and tackle important societal issues.

Complete solution to open problems on exponential augmented Zagreb index of chemical trees

One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G asEAZ(G)=∑vivj∈E(G)e(didjdi+dj−2)3, where di signifies the degree of vertex vi, and E(G) is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.

Topological Characterization for Triangular, RegularTriangular Oxides and Silicates Networks

Chemical graph theory can be studied with the aid of mathematical tools called m-polynomials. M-Polynomials offer a potent tool for computing different topological indices associated with vertex degrees and analyzing degree-based structural information in graphs. By counting specific substructure types within them, they are able to encode information about the structure of molecules or networks. In this article, we have developed M-Polynomials with the help of different topological invariants such as first Zagreb (M1(β)), second Zagreb (M2(β)), second modified Zagreb (Mm2(β)), inverse sum (I(β)), harmonic index (H(β)) and Randic index (Rα0(β)) for the molecular structures of Triangular oxide TOX(r), Regular triangular oxide RTOX(r), Triangularsilicate TSL(r) & Regular triangular silicate RTSL(r) networks to introduce new closed formulas to get better understanding the applications of M-Polynomials and topological indices in mathematical chemistry especially in the field of QSAR and QSPR study with the help of some software like MATLAB. We have also discussed the graphical behaviors of the above-mentioned structures.

Extremal graphs with given parameters in respect of general ABS index

For a graph G=(V,E), the general atom-bond sum-connectivity index is formulated by ABSσ(G)=∑ξζ∈E(G)(dξ+dζ−2dξ+dζ)σ, where dξ indicates the degree of vertex ξ∈V, σ can be arbitrary real number. Several physicochemical features of benzenoid hydrocarbons can be correctly anticipated using the ABSσ index. Researchers go through careful inspection of some ABSσ and reveal that ABS1, ABS−1, ABS12, ABS3 enjoy benefit in foreseeing the boiling point, the standard enthalpy of vaporization, the acentric factor, and the entropy of octane isomers, separately. One significant wellspring of information for exploring the molecular construction is the assessed worth of the reach for topological indices through certain predefined molecular graph parameters. The purpose of the present work is to find the highest values of the ABSσ index for σ≥0 in the set of graphs that were given certain predefined parameters, such as matching number, chromatic number, etc. Furthermore, illustrations of the relevant extremal graphs were provided.

Spanning Trees Whose Stems are Caterpillars

Let 𝑇 be a tree, a vertex of degree one is called a leaf. The set of all leaves of 𝑇 is denoted by Leaf(𝑇). The subtree 𝑇 − Leaf(𝑇) of 𝑇 is called the stem of 𝑇 and denoted by Stem(𝑇). A tree 𝑇 is called a caterpillar if Stem(𝑇) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.

Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs

Extended Reverse R Degrees of Vertices and Extended Reverse R indices of Graphs

Approximating power node-deletion problems

In the Power Vertex Cover(PVC) problem introduced in [1] as a generalization of the well-known Vertex Cover, we are allowed to specify costs for covering edges in a graph individually. Namely, two (nonnegative) weights, w(u,v) and w(v,u), are associated with each edge {u,v}∈E of an input graph G=(V,E), and to cover an edge {u,v}, it is required to assign “power” p∈R+V on vertices of G such that either p(u)≥w(u,v) or p(v)≥w(v,u). The objective is to minimize the total power assigned on V, ∑v∈Vp(v), while covering all the edges of G by p.The node-deletion problem for a graph property π is the problem of computing a minimum vertex subset C⊆V in a given graph G=(V,E), such that the graph satisfies π when all the vertices in C are removed from G. In this paper we consider node-deletion problems extended with the “covering-by-power” condition as in PVC, and present a unified approach for effectively approximating them. The node-deletion problems considered are Partial Vertex Cover (PartVC), Bounded Degree Deletion (BDD), and Feedback Vertex Set (FVS), each corresponding to graph properties π= “the graph has at most |E|−k edges”, π= “vertex degree of v is no larger than b(v)”, and π= “the graph is acyclic”, respectively. After reducing these problems to the Submodular Set Cover (SSC) problem, we conduct an extended analysis of the approximability of these problems in the new setting of power covering by applying some of the existing techniques for approximating SSC. It will be shown that 1) PPartVC can be approximated within a factor of 2, 2) PBDD for b∈Z+V within max⁡{2,1+bmax}, where bmax=maxv∈V⁡b(v), or within 2+log⁡bmax (for bmax≥1) by a combination of the greedy SSC algorithm and the local ratio method extended for power node-deletion problems, and 3) PFVS within 2, resulting in each of these bounds matching the best one known for the corresponding original problem.

Thin edges in cubic braces

AbstractFor a vertex set in a graph, the edge cut is the set of edges with exactly one end vertex in . An edge cut is tight if every perfect matching of the graph contains exactly one edge in . A matching covered bipartite graph is a brace if, for every tight cut , or , where . Braces play an important role in Lovász's tight cut decomposition of matching covered graphs. The bicontraction of a vertex of degree two in a graph, with precisely two neighbours and , consists of shrinking the set to a single vertex. The retract of a matching covered graph is the graph obtained from by repeatedly the bicontractions of vertices of degree two. An edge of a brace with at least six vertices is thin if the retract of is a brace. McCuaig showed that every brace of order at least six has a thin edge. In a brace of order six or more, Carvalho, Lucchesi and Murty proved that has two thin edges, and conjectured that contains two nonadjacent thin edges. Further, they made a stronger conjecture: There exists a positive constant such that every brace has thin edges. By showing that, in every cubic brace of order at least six, there exists a matching of size at least such that every edge in is thin, we prove that the above two conjectures hold for cubic braces.

On tricyclic graphs with maximum atom–bond sum–connectivity index

The sum-connectivity, Randić, and atom-bond connectivity indices have a prominent place among those topological indices that depend on the graph's vertex degrees. The ABS (atom-bond sum-connectivity) index is a variant of all the aforementioned three indices, which was recently put forward. Let T(n) be the class of all connected tricyclic graphs of order n. Recently, the problem of determining graphs from T(n) having the least possible value of the ABS index was solved in (Zuo et al., 2024 [39]) for the case when the maximum degree of the considered graphs does not exceed 4. The present paper addresses the problem of finding graphs from T(n) having the largest possible value of the ABS index for n≥5.

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.

Construction of S(3)(2,3)-Designs of Any Index

Let H(3) be a uniform hypergraph of rank 3. A hyperstar S(3)(2,3) of centreC={x,y} is a 3-uniform hypergraph with three hyperedges, all having the centreC={x,y} in common, with x and y of degree 3 and the remaining vertices of degree 1. In this paper, we determine the spectrum of S(3)(2,3)-designs for any index λ.

KEPLER BANHATTI AND MODIFIED KEPLER BANHATTI INDICES

We introduce a novel vertex degree based topological index, called Kepler Banhatti index. Also we put forward the modified Kepler Banhatti index of a graph. We propose the Kepler Banhatti and modified Kepler Banhatti exponentials of a graph. In this study, we determine the newly defined the Kepler Banhatti indices and their corresponding exponentials for certain dendrimers. Furthermore, we establish some properties of the Kepler Banhatti index.

On the maximum atom-bond sum-connectivity index of molecular trees

Let G be a graph with V(G) and E(G), as vertex set and edge set, respectively. The atom-bond sum-connectivity (ABS) index is a vertex-based topological index which is defined as ABS ( G ) = ∑ ab ∈ E ( G ) ϱ a + ϱ b − 2 ϱ a + ϱ b , where ϱ a is the degree of the vertex a. In this paper, we obtain sharp upper bounds for the ABS index of molecular trees in terms of order and number of branching vertices and vertices of degree two.

Cairo lattice with time-reversal non-invariant vertex couplings

We analyze the spectrum of a periodic quantum graph of the Cairo lattice form. The used vertex coupling violates the time reversal invariance and its high-energy behavior depends on the vertex degree parity; in the considered example both odd and even parities are involved. The presence of the former implies that the spectrum is dominated by gaps. In addition, we discuss two modifications of the model in which this is not the case, the zero limit of the length parameter in the coupling, and the sign switch of the coupling matrix at the vertices of degree three; while different they both yield the same probability that a randomly chosen positive energy lies in the spectrum.

On the rank, Kernel, and core of sparse random graphs

AbstractWe study the rank of the adjacency matrix of a random Erdős‐Rényi graph . It is well known that when , with high probability, is singular. We prove that when , with high probability, the corank of is equal to the number of isolated vertices remaining in after the Karp‐Sipser leaf‐removal process, which removes vertices of degree one and their unique neighbor. We prove a similar result for the random matrix , where all entries are independent Bernoulli random variables with parameter . Namely, we show that if is the bipartite graph with bi‐adjacency matrix , then the corank of is with high probability equal to the max of the number of left isolated vertices and the number of right isolated vertices remaining after the Karp‐Sipser leaf‐removal process on . Additionally, we show that with high probability, the ‐core of is full rank for any and . This partially resolves a conjecture of Van Vu for . Finally, we give an application of the techniques in this paper to gradient coding, a problem in distributed computing.

Structure of some ( [formula omitted], [formula omitted])-free graphs with application to colorings

We use Pt and Ct to denote a path and a cycle on t vertices, respectively. A diamond is a graph obtained from two triangles that share exactly one edge. A kite is a graph that consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A gem is a graph that consists of a P4 plus a vertex adjacent to all vertices of the P4. Choudum et al. (2021) proved that every (P7,C7,C4, gem)-free graph has either a clique cutset or a vertex whose neighbors is the union of two cliques unless it is a clique blowup of the Petersen graph, and every non-Petersen (P7,C7,C4, diamond)-free graph has either a clique cutset or has a vertex of degree at most max{2,ω(G)−1}. In this paper, we extend these two results respectively to (P7,C4, gem)-free graphs and (P7,C4, diamond)-free graphs. We also prove that if G is a (P7,C4, kite)-free graph with δ(G)≤ω(G) and without clique cutsets, then there is an integer ℓ≥0 such that G=Kℓ+H, where H is either the Petersen graph or a well-defined graph F. This implies a polynomial algorithm which can determine the chromatic number of (P7,C4, diamond)-free graphs, color (P7,C4, kite)-free graphs G with (ω(G)+1) colors, and color (P7,C4, gem)-free graphs G with (2ω(G)−1) colors.

Quasi-Laplacian energy of fractal graphs

Graph energy is a measurement of determining the structural information content of graphs. The first Zagreb index can be handled with its connection to graph energy. In this paper, a novel and significant application of the first Zagreb index to composite graphs based on fractal graphs is revealed, and by the relation between quasi-Laplacian energy and the vertex degrees of a graph, we derive closed-form formulas for the quasi-Laplacian energy of fractal graphs or namely R-graphs, R-vertex and edge join, R-vertex and edge corona, R-vertex and edge neighborhood graphs in terms of the corresponding energy, the first Zagreb indices, number of vertices and edges of the underlying graphs.

On the maximum local mean order of sub‐k $k$‐trees of a k $k$‐tree

AbstractFor a ‐tree , a generalization of a tree, the local mean order of sub‐‐trees of is the average order of sub‐‐trees of containing a given ‐clique. The problem whether the maximum local mean order of a tree (i.e., a 1‐tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any ‐tree , does the maximum local mean order of sub‐‐trees containing a given ‐clique occur at a ‐clique that is not a major ‐clique of ? In this paper, we give it an affirmative answer.

First and second K Banhatti indices of a fuzzy graph applied to decision making

Topological index of a graph is a numerical representation linked to the graph or the molecular structure associated with the graph, shedding light on its topological and structural properties. Different topological indices showcase unique attributes rooted in their degree, spectrum, and distance characteristics. Among these, the first and second K Banhatti indices, introduced by Kulli, involving both vertex and edge degrees, have garnered significant scholarly interest, prompting extensive discussions in graph theory. Within this research paper, a thorough exploration delves into the attributes surrounding the fuzzified first and second K Banhatti indices. This exploration is directed towards several bounds of the index and index of various structures like cycles, complete fuzzy graphs, complete bipartite graphs and stars. Relation between a fuzzy graph and its partial subgraphs, fuzzy graph and its complement graph in connection with the indices is also found. Additionally, the research paper investigates the first and second K Banhatti indices concerning graph operations such as the corona product and coalescence. Additionally, the paper outlines an algorithm designed to accurately compute the first and second K Banhatti indices and proposes an application in decision-making.