Family Of Graphs Research Articles

On topological indices of certain families of graphs

Abstract The aim of this paper is to compute topological indices such as general randic index,
general sum connectivity index, atom bond connectivity index, geometric arithmetic
index, forgotten index, rst zagreb index, second zagreb index, rst multiple zagreb
index, second multiple zagreb index and hyper zagreb index of di erent families of
graphs.

A Laplacian eigenbasis for threshold graphs

Abstract Let G G be a graph on n n vertices. In this article, we prove that an eigenbasis of the Laplacian matrix of a star graph of order n n is also an eigenbasis of G G if and only if G G is a threshold graph. As an application of this spectral characterization, we show an infinite family of threshold graphs that are weakly Hadamard diagonalizable.

Graphs with girth 2ℓ+1 $2\ell +1$ and without longer odd holes are 3‐colorable

AbstractFor a number , let denote the family of graphs which have girth and have no odd hole with length greater than . Wu et al. conjectured that every graph in is 3‐colorable. Chudnovsky et al. and Wu et al., respectively, proved that every graph in and is 3‐colorable. In this paper, we prove that every graph in is 3‐colorable.

Error and attack vulnerability of Apollonian networks

Abstract This article examines the resilience of different Apollonian network (AN) types—deterministic, random, and evolutionary—to systematic attacks. ANs, members of the family of maximal planar graphs, possess unique properties such as high clustering coefficients, small-world properties, scale-free behavior, Euclidean and space-filling properties, and modularity. These peculiarities require a thorough investigation of their robustness. This work presents a novel approach to studying ANs by implementing evolutionary Apollonian networks (EANs). These EANs include various probabilities distribution functions, including exponential, degenerate, logistic, Pareto, and stable (Cauchy, Lévy, Normal) distributions. To improve the robustness of these networks, we propose a novel edge rewiring mechanism using a genetic algorithm (GA). The GA aims to optimize a combined metric that includes the Flow Robustness of Degree (SFRD), Betweenness (SFRB), and Dangalchev's closeness (SFRC) centralities while preserving the original degree distribution and structural properties of the network. To evaluate the effectiveness of this approach, we use various robustness measures to assess the resilience of different AN types. The results show that SFRB, SFRD, and SFRC effectively rank ANs based on their robustness.

Weak Randomness in Graphons and Theons

ABSTRACTCall a hereditary family of graphs strongly persistent if there exists a graphon such that in all subgraphons of , is precisely the class of finite graphs that have positive density in . Our first result is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. We call graphons with the self‐similarity property above weakly random. A hereditary family is said to have the weakly random Erdős–Hajnal property () if every graphon that is a limit of graphs in has a weakly random subgraphon. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to as those with “few” prime graphs. We also extend some of the results above to structures in finite relational languages by using the theory of theons.

The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

Upper Bounds for Linear Graph Codes

ABSTRACTA linear graph code is a family of graphs on vertices with the property that the symmetric difference of the edge sets of any two graphs in is also the edge set of a graph in . In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph . In particular, we show that if has an even number of edges, the size of the code is , making progress on a question of Alon. Furthermore, we show that for almost all graphs with an even number of edges, there exists such that the size of a linear graph code without a copy of is at most .

ON THE EVANS CHAIN COMPLEX

Abstract We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$ -algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$ -algebras with trivial K-theory. Additionally, in the specialised case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

Results on super edge magic deficiency of somewell-known classes of finite graphs

A graph Ω(Λ, Γ) is considered super edge magic if there exists a bijective function ϕ:Λ(Ω)∪Γ(Ω)→ {1, 2, 3, …,|Λ(Ω)|+|Γ(Ω)|} such that ϕ(τ1)+(τ1τ2)+ ϕ(τ2) is a constant for every edge τ1τ2 ∈ Γ(Ω), and ϕ(Λ(Ω))= {1, 2, 3, …,|Λ(Ω)|}. Furthermore, the super edge magic deficiency of a graph Ω, denoted as μs (Ω), is either the minimum non-negative integer η such that Ω∪ηK1 is a super edge magic graph or +∞ if such an integer η does not exist. In this paper, we investigate the super edge magic deficiency of certain families of graphs.

Characterizations of kites as graceful graphs

We introduce and study an infinite family of graceful graphs, which we call kites. The kites are graphs where a path is joined with a graph "forming" a kite. We study and characterize three classes of the kites: kites formed by cycles known to be graceful, fan kites and lantern kites. Beside showing in a transparent way that all these graphs are graceful, we provide characterizations of these graphs among all simple graphs via three tools: via Sheppard's labelling sequences introduced in the 1970s and via labelling relations and graph chessboards. The latter are relatively new tools for the study of graceful graphs introduced by Haviar and Iva\v ska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a~nice visualization of the graceful labellings.

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

Positive semidefiniteness of $$D_{\alpha }$$-matrix on some families of graphs

Positive semidefiniteness of $$D_{\alpha }$$-matrix on some families of graphs

Chromatic Coloring of Distance Graphs V

Let G = ( V , E ) be any graph. If there exists an injection f : V → Z , such that | f ( u ) – f ( v ) | is prime for every u v ∈ E , then we say G is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of n mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of n arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

Well-indumatched Pseudoforests

A graph is called well-indumatched if all of its maximal induced matchings have the same size. Akbari et al. [1] provided a characterization of well-indumatched trees and some results on well-indumatched unicyclic graphs. In this paper, we extend this result by providing a complete structural characterization of well-indumatched pseudotrees, in which each component has no cycle or a unique cycle, by identifying the well-indumatched graph families.

On a family of quasi-strongly regular Cayley graphs

In this paper, we construct a family of quasi-strongly regular Cayley graphs which is defined on a finite group G with respect to a proper subgroup H of G and denoted by Γ H ( G ) . We also compute the full automorphism group and characterize various transitivity properties of Γ H ( G ) for any group G and for any subgroup H of G.

The List-Ramsey threshold for families of graphs

AbstractGiven a family of graphs $\mathcal{F}$ and an integer $r$ , we say that a graph is $r$ -Ramsey for $\mathcal{F}$ if any $r$ -colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$ . The threshold for the classic Ramsey property, where $\mathcal{F}$ consists of one graph, in the binomial random graph was located in the celebrated work of Rödl and Ruciński.In this paper, we offer a twofold generalisation to the Rödl–Ruciński theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$ , where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa–Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$ -statement of the Rödl–Ruciński theorem.

E-Cordial Labeling of Some Families of Graphs

An E-cordial labeling σ: E →{0,1} induces σ∗: V →{0,1} on graph G=(V,E), where (σ(v)=(∑_(u∈V)▒〖σ(uv)〗) mod 2 is taken over all edges uv∈E, and the labelling satisfies the conditions |vσ (0)-vσ (1)|≤1, |eσ (0)-eσ (1)|≤1. Where vσ (k) represent the count of vertices within the graph G that bear the label k based on the labeling function σ (here k can be either 0 or 1). Similarly, eσ∗ (k) denotes the quantity of edges in the graph G that link vertices labeled with the value k according to the labeling function σ∗ (here k can be 0 or 1). A graph along with E-cordial labeling is called an E-cordial graph. We prove that the graphs such as Herschel graph H, Durer graph, Frucht graph, Tietze graph, hypohamiltonian graph, truncated tetrahedron graph, cubic graph with 12 vertices, Wagner graph, Moser spindle graph, Goldner-Harary graph and diamond graph are E-cordial graphs.

A family of graphs that cannot occur as character degree graphs of solvable groups

A family of graphs that cannot occur as character degree graphs of solvable groups

Type-II Apollonian network: More robust and more efficient Apollonian network

The family of planar graphs is a particularly important family and models many networks including the layout of printed circuits. The widely-known Apollonian packing process has been used as guideline to create the typical Apollonian network with planarity. In this paper, we propose a new principled framework based on the Apollonian packing process to generate model as complex network, and obtain a family of new networks called Type-II Apollonian network At. While our network and the typical Apollonian network are maximal planar, the former turns out to be Hamiltonian and Eulerian, however, the latter is not. Then, we in-depth study some fundamental structural properties on network At, and verify that network At is sparse, has scale-free feature and small-world property, and exhibits disassortative mixing structure. Next, we derive the asymptotic solution of the spanning tree entropy of network At by designing an effective algorithm, which suggests that Type-II Apollonian network is more robust to a random removal of edges than the typical Apollonian network. Additionally, we study trapping problem on network At, and use average trapping time as metric to show that Type-II Apollonian network At has more efficient underlying structure for fast information diffusion than the typical Apollonian network.

Splitting Plane Graphs to Outerplanarity

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is NP-hard.Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove NP-completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Additionally, we show upper and lower bounds for certain families of maximal planar graphs. Finally, we provide a SAT formulation for the problem, and evaluate it on a small benchmark.