Family Of Graphs Research Articles

On the minimum size of graphs with given generalized connectivity

Let G be a connected graph and S⊆V(G) with |S|≥2. A tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are internally disjoint if E(T1)∩E(T2)=0̸ and V(T1)∩V(T2)=S. For an integer k≥2, the generalizedk-connectivity of a graph G, denoted by κk(G), is defined as κk(G)=min{κG(S):S⊆V(G) and |S|=k}, where κG(S) denotes the maximum number of pairwise internally disjoint S-trees in G. The generalized connectivity is a generalization of traditional connectivity. Let Gn be the class of connected graphs of order n and let f(n,k,t)=minG∈Gn{|E(G)|:κk(G)=t}. In this paper, we prove f(n,k,t)≥⌈t(t+2)2t+2n⌉ for 4≤k≤n and 1≤t≤n−⌈k2⌉. In particular, the lower bound is sharp when k=4 and t=2 (i.e., we explicitly provide a family of graphs fulfilling the bound in this case), improving the known results of Sun et al. (2021).

The oriented chromatic number of the hexagonal grid is 6

The oriented chromatic number of a directed graph G is the minimum order of an oriented graph to which G has a homomorphism. The oriented chromatic number χo(F) of a graph family F is the maximum oriented chromatic number over any orientation of any graph in F. For the family of hexagonal grids H2, Bielak (2006) proved that 5≤χo(H2)≤6. Here we close the gap by showing that χo(H2)≥6.

Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees

The planar slope numberpsn(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{psn}\\,}}(G)$$\\end{document} of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that psn(G)∈O(cΔ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{psn}\\,}}(G) \\in O(c^{\\Delta })$$\\end{document} for every planar graph G of maximum degree Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}. This upper bound has been improved to O(Δ5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\Delta ^5)$$\\end{document} if G has treewidth three, and to O(Δ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\Delta )$$\\end{document} if G has treewidth two. In this paper we prove psn(G)≤max{4,Δ}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{psn}\\,}}(G) \\le \\max \\{4,\\Delta \\}$$\\end{document} when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that O(Δ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\Delta ^2)$$\\end{document} slopes suffice for nested pseudotrees.

Computing The Energy of Certain Graphs based on Vertex Status.

The concept of Hückel molecular orbital theory is used to compute the graph energy numerically as well asnd graphically, on the base of the status of a vertex. Our aim is to explore the graph energy of various graph families on the base of the status adjacency matirx matrix and its Laplacian version. We opt for the technique of finding eigenvalues of adjacency and Laplacian matrices constructed on the base of the status of vertices. We explore the exact status sum and Laplacian status sum energies of a complete graph, complete bipartite graph, star graphs, bistar graphs, barbell graphs and graphs of two thorny rings. We also compared the obtained results of energy numerically and graphically. In this article, we extended the study of graph spectrum and energy by introducing the new concept of the status sum adjacency matrix and the Laplacian status sum adjacency matrix of a graph. We investigated and visualized these newly defined spectrums and energies of well-known graphs, such as complete graphs, complete bi-graphs, star graphs, friendship graphs, bistar graphs, barbell graphs, and thorny graphs with 3 and 4 cycles.

Isospectral graphs via spectral bracketing

In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenfunctions of the corresponding graphs. In principle, one only needs to know the multiplicity of the eigenvalues of the auxiliary graphs, and that these eigenvalues are all different. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy balls, complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.

Graphs obtained by disjoint unions and joins of cliques and stable sets

We consider the set of graphs that can be constructed from a one-vertex graph by repeatedly adding a clique or a stable set linked to all or none of the vertices added in previous steps. This class of graphs contains various well-studied graph families such as threshold, domishold, co-domishold and complete multipartite graphs, as well as graphs with linear clique-width at most 2. We show that it can be characterized by three forbidden induced subgraphs as well as by properties involving maximal stable sets and minimal dominating sets. We also give a simple recognition algorithm and formulas for the computation of the stability and domination numbers of these graphs.

Legal Hop Independent Sequences in Graphs

Let G be any graph. A sequence L = (w1, w2, ..., wk) of distinct vertices of G is called a legal sequence if NG[wi]\i[−1j=1NG[wj ] ̸= ∅ for each i ∈ {2, ..., k}. A legal sequence L is called a legal hop independent sequence if dG(wi, wj ) ̸= 2 for each i ̸= j. The maximum length of a legalhop independent sequence in G, denoted by αℓh(G), is called the legal hop independence number of G. In this paper, we initiate the study of legal hop independent sequences in a graph. Weinvestigate its relationships with the hop independence and grundy domination parameter of a graph, respectively. In fact, the legal hop independence parameter is at most equal to the grundy domination (resp. hop independence) parameter on any graph G. Moreover, we derive some formulas or bounds of this parameter on some families of graphs, join, and corona of two graphs.

Topological numbers of fuzzy soft graphs and their applications in globalizing the world by mutual trade

In this research draft, by keeping in view, the importance of Zagreb numbers, first we defined some familiar graph families and their degrees in fuzzy soft environment, and then by converting these two novel topological numbers in fuzzy soft framework, their formulae are calculated for these graphical networks. An application of international trade using two kinds of routes is also discussed in the article, and by comparing the results efficiency of both the numbers is compared and analyzed how topological descriptors are useful in examining the progress and profit in the countries, that are a part of the project. Graphs are used in the modeling of links and processes within physical, biological, social, and information systems. A fuzzy graph is the most adaptable tool in mathematics, which describes the fuzzy relationship between objects. Fuzzy graphs are very helpful to give more accuracy to the system as compared to crisp graphs. A classification of the universe with respect to certain specified parameters is known as a soft set. It is a new method for simulating ambiguity and uncertainty. Soft set theory was created by Molodtsov in 1999 as a method for simulating vagueness and uncertainty. Soft set theory is presently used widely in academia to address decision-making issues. The concept of soft graphs is used to provide a parameterized point of view for graphs. In many areas of mathematics and chemistry, as well as in network theory, spectral graph theory, and molecular chemistry, the Zagreb numbers are crucial graph parameters. Due to the importance and significance of Zagreb numbers, in this research article we define these graph parameters in fuzzy soft environment and then derived the fundamental formula for first and second Zagreb numbers of some popular graph families. Also by keeping in view the importance of graph theory in solving decision making problems in business industry, and the most generic behavior of fuzzy soft graph due to its parameterized nature we use our work in application of decision making in trade among different nations.

INDEPENDENT AVERAGE CONNECTIVITY NUMBER IN GRAPHS

Let G = (V (G), E(G)) be a simple, undirected graph. The vulnerability of a graph is a determination that includes certain properties of the graph not to be damaged after the removal of a number of vertices or edges. In this paper, the concept of independent average connectivity number is introduced as a new vulnerability measure and some bounds and independent average connectivity number of some well-known graph families are examined. Independent average connectivity will give the number of vertices that need to be removed to separate the pairs of vertices selected from the independent set in a graph G(V, E). Independent average connectivity number of a graph G(V, E) is denoted by kβ(G) and defined as kβ(G) = T I(G) ( n 2 ) where T I(G) is a total independence number of G(V, E) graph with n vertices. In this study, all the graphs considered simple, finite, and undirected.

On r-dynamic k-coloring of ladder graph families

The concept of [Formula: see text]-dynamic [Formula: see text]-coloring of a graph [Formula: see text] refers to a proper vertex [Formula: see text]-coloring where the number of colors assigned to the vertices in the neighborhood of each vertex [Formula: see text] is greater than or equal to the min [Formula: see text]. This study focuses on the [Formula: see text]-dynamic coloring of various types of ladder graphs, which are derived from the Cartesian product of [Formula: see text] and [Formula: see text], including the Diagonal Ladder Graph, Open Diagonal Ladder Graph, Triangular Ladder Graph, Open Triangular Ladder Graph and Slanting Ladder Graph.

Optimal Layout of (Kp − Cp)n into Wheel-Like Networks

The problem of finding the induced subgraph with the maximum number of edges among all subsets of a fixed cardinality is known as the maximum subgraph problem (MSP). The lexicographic order has been proven optimal for Cartesian products of specific regular graphs [B. Sergei, B. Pavle and K. Nikola, New infinite family of regular edge isoperimetric graphs, Theoretical Computer Science 721 (2018) 42–53]. This paper is dedicated to the precise calculation of edge values within the resulting subgraphs, enhancing our comprehension of their structural properties and implications. In this paper, we focus on solving the MSP of [Formula: see text] for any [Formula: see text] and for odd [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is obtained by the Cartesian product of a complete graph with [Formula: see text] vertices with a removal of a cycle on [Formula: see text] vertices. It has been observed that the graph obtained by deleting a cycle of length [Formula: see text] from a complete graph [Formula: see text] is isomorphic to a circulant graph [Formula: see text], while the particular case [Formula: see text] has been previously studied in [A. Syeda and M. Rajesh, MinLA of [Formula: see text] and its optimal layout into certain trees, The Journal of Supercomputing (2023), https://doi.org/10.1007/s11227-023-05140-3]. Our work extends the understanding of the MSP to a broader class of graphs with similar properties and applications.

On the parameterized complexity of freezing dynamics

In this paper we establish how alphabet size, treewidth and maximum degree of the underlying graph are key parameters which influence the overall computational complexity of finite freezing automata networks. First, we define a general decision problem, called Specification Checking Problem, that captures many classical decision problems such as prediction, nilpotency, predecessor, asynchronous reachability.Then, we present a fast-parallel algorithm that solves the general model checking problem when the three parameters are bounded, hence showing that the problem is in NC. Moreover, we show that the problem is in XP on the parameters tree-width and maximum degree.Finally, we show that these problems are hard from two different perspectives. First, the general problem is W[2]-hard when taking either treewidth or alphabet as single parameter and fixing the others. Second, the classical problems are hard in their respective classes when restricted to families of graph with sufficiently large treewidth.

Spread of influence with incentives in edge-weighted graphs with emphasis on some families of graphs

Spread of influence with incentives in edge-weighted graphs with emphasis on some families of graphs

Graphs with large multiplicity of −2 in the spectrum of the eccentricity matrix

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by keeping only the entries that are largest in at least one of their row or column. This matrix can be seen as a counterpart to the standard adjacency matrix, since the latter one is also obtained from the distance matrix but this time by keeping only the entries equal to 1. It is known that, for λ∉{−1,0} and a fixed i∈N, when n passes N there is only a finite number of graphs with n vertices having λ as an eigenvalue of multiplicity n−i in the spectrum of the adjacency matrix. This phenomenon motivates us to consider graphs with large multiplicity of an eigenvalue of the eccentricity matrix. In this context, we determine all connected graphs with n vertices for which −2 has the multiplicity n−i, where i≤5. Infinite families of graphs with this spectral property are encountered. Results of this paper can be compared to results concerning graphs with large multiplicity of zero in the spectrum of the adjacency matrix; at present the graphs for which this multiplicity is n−i, where i≤4, are known. Our results also become meaningful in the framework of the median eigenvalue problem since, for sufficiently large n, the median eigenvalue of the obtained graphs is always −2.

Indicated coloring of some families of graphs

Indicated coloring of some families of graphs

Local Multiset Dimension of Amalgamation Graphs.

Background: One of the topics of distance in graphs is the resolving set problem. Suppose the set W = { s 1, s 2, …, s k} ⊂ V ( G), the vertex representations of ∈ V ( G) is r m( x| W) = { d( x, s 1), d( x, s 2), …, d( x, s k)}, where d( x, s i) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicity. The set W is called a local m-resolving set of graphs G if r m( v| W)≠ r m( u| W) for uv ∈ E( G). The local m-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by md l( G). Thus, if G has an infinite local multiset dimension and then we write . Methods: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set W ⊂ V ( G); determine the vertex representation of two adjacent vertices in G; and prove the theorem. Results: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely md l( Amal( G, v, m)) ≤ m. md l( G) and their exact value of local multiset dimension of some families of graphs namely md l( Amal( P n, v, m)) = 1, , md l( Amal( W n, v, m)) = m. md l( W n), md l( Amal( F n, v, m)) = m. md l( F n) for d( v) = n, . Conclusions: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs.

Curvature Sets Over Persistence Diagrams

We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers k≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 0$$\\end{document} and n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 1$$\\end{document} we consider the dimension k Vietoris–Rips persistence diagrams of all subsets of a given metric space with cardinality at most n. We call these invariants persistence sets and denote them as Dn,kVR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{n,k}^{\ extrm{VR}}$$\\end{document}. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parameters n and k, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which D4,1VR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{D}}_{4,1}^{\ extrm{VR}}$$\\end{document} fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a space X with cardinality 2k+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+2$$\\end{document} with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction.

Bipartite Unique Neighbour Expanders via Ramanujan Graphs.

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to being implementable in practice, due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalises the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction, we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalises Kahale's average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.

Forbidden family of Ph-magic graphs

Forbidden family of Ph-magic graphs

Graphs with girth 9 and without longer odd holes are 3‐colourable

AbstractFor a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu, Xu and Xu conjectured that every graph in is 3‐colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in , and is 3‐colourable, respectively. In this paper, we prove that every graph in is 3‐colourable. This confirms Wu, Xu and Xu's conjecture.