Graphs G1 Research Articles

Extra connectivity of networks modeled by the strong product graphs

Given a non-negative integer g, the g-extraconnectivity in a connected graph G is defined as the minimum number of vertices that, when removed, results in G becoming disconnected and each remaining component containing more than g vertices. As an extension of the connectivity, the g-extraconnectivity provides a better measurement for the fault tolerance of an interconnection network. The strong product G1⊠G2 of graphs G1 and G2 is the graph with vertex set V(G1)×V(G2), where two distinct vertices (a1,b1) and (a2,b2) are adjacent if and only if a1=a2 and b1b2∈E(G2), or b1=b2 and a1a2∈E(G1), or a1a2∈E(G1) and b1b2∈E(G2). In this paper, we focus on networks modeled by the strongproductG1⊠G2. We determine the g(≤3)-extraconnectivity of G1⊠G2, where G1 and G2 are regular and maximally connected graphs with girth at least g+4. Additionally, we give the g(≤3)-extra conditional diagnosability of G1⊠G2 under PMC model.

A bound on relative lengths of triangle-free graphs

For a 2-connected graph G, the relative length of G, denoted by diff(G), is the difference between the orders of a longest path and a longest cycle in G. This parameter is used as a measure to estimate how close a given graph is to a hamiltonian graph. Let σk(G) be the least value of the sums of degrees of vertices in independent sets of cardinality k. In 2008, Paulusma and Yoshimoto proved that a 2-connected triangle-free graph G of order n with σ4(G)≥n+2 satisfies diff(G)≤1 unless G is isomorphic to one exceptional graph G0. In this paper, we extend their result and prove that for an integer s with 23(n+4)<s≤n+2, a 2-connected triangle-free graph of order n with σ4(G)≥s satisfies diff(G)≤n+3−s unless s=σ4(G)=n+2 and G=G0.

Fair Detour Domination in the Corona of Two Graphs: Optimizing CCTV Camera Installation for Efficient Surveillance

The objective of our research is to analyze the properties of FDD sets in the corona of two graphs and explore their practical application in CCTV camera installation. A set F ⊆ V(G) is considered to be a Fair Detour Dominating set (FDD-set) if it is detour dominating and the number of neighbors within set F is the same for any pair of vertices outside of F. Among these FDD sets, the fγd-set refers to the FDD-set with the smallest number of vertices, and its order defines the Fair Detour Domination number (fγd (G)). We have established that for any arbitrary graphs G1 and G2, fγd (G1 o G2) = |V(G1) iff fd(G2) = 1 we have determined the fγd number of corona products of any connected graph G with the path graph as well as the cycle graph. We also characterized FDD sets in the corona product of two connected graphs and provided a thorough description of how FDD sets can be used in optimizing CCTV camera installation for efficient surveillance.

Some extremal problems on [formula omitted]-spectral radius of graphs with given size

Nikiforov defined the Aα-matrix of a graph G as Aα(G)=αD(G)+(1−α)A(G), where α∈[0,1], D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of Aα(G) is called the Aα-spectral radius of G, denoted by ρα(G). In this paper, we first give an upper bound on ρα(G) of a connected graph G with fixed size m≥3k and maximum degree Δ≤m−k, where k is a positive integer. For two connected graphs G1 and G2 with size m≥4, employing this upper bound, we prove that ρα(G1)>ρα(G2) if Δ(G1)>Δ(G2) and Δ(G1)≥2m3+1. As an application, we determine the graph with the maximal Aα-spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph.

Graph-codes

The symmetric difference of two graphs G1,G2 on the same set of vertices [n]={1,2,…,n} is the graph on [n] whose set of edges are all edges that belong to exactly one of the two graphs G1,G2. Let H be a fixed graph with an even (positive) number of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs on [n] containing no two members whose symmetric difference is a copy of H. Is it true that DH(n)=o(2n2) for any such H? We discuss this problem, compute the value of DH(n) up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.

On the cycle-path bipartite Ramsey number

For bipartite graphs G1,G2,…,Gk, the bipartite Ramsey number b(G1,G2, …,Gk) is the least positive integer b, so that any coloring of the edges of Kb,b with k colors, will result in a copy of Gi in the ith color, for some i. In this paper we will show that if s is an integer such that s≥max⁡{18,(b(C34,C34)+1)/2}, then b(C2s,P2s)=2s−1.

On the achromatic number of the Cartesian product of two complete graphs

A vertex colouring f: V(G) → C of a graph G is complete if for any c1, c2 ∈ C with c1 ≠ c2 there are in G adjacent vertices v1, v2 such that f(v1) = c1 and f(v2) = c2. The achromatic number of G is the maximum number achr(G) of colours in a proper complete vertex colouring of G. Let G1▫G2 denote the Cartesian product of graphs G1 and G2. In the paper achr(Kr2 + r + 1▫Kq) is determined for an infinite number of q’s provided that r is a finite projective plane order.

The potential-Ramsey numbers [formula omitted] and [formula omitted

A nonincreasing sequence ρ=(ρ1,…,ρn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ρ. Given two graphs G1 and G2, Busch et al. introduced the potential-Ramsey number of G1 and G2, denoted rpot(G1,G2), is the smallest nonnegative integer m such that for every m-term graphic sequence ρ, there is a realization G of ρ with G1⊆G or with G2⊆G¯, where G¯ is the complement of G. For t≥2 and 0≤k≤⌊t2⌋, let Kt−k be the graph obtained from Kt by deleting k independent edges. Busch et al. determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=0. Du and Yin determined rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for k=1. In this paper, we further determine rpot(Cn,Kt−k) and rpot(Pn,Kt−k) for 2≤k≤⌊t2⌋.

Constant Time Calculation of the Metric Dimension of the Join of Path Graphs

The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,⋯,wk of V(G) is a resolving set for G, if for ∀u,v∈V(G), there exists wi∈W ∋ d(u,wi)≠d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1∪V2 and the edge set is E=E1∪E2∪uv,u∈V1,v∈V2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.

Transitive closure in a polluted environment

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph G0 of initially occupied edges on n vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When G0 is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where G0 is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is (logn)−1/2+o(1). We conclude with several conjectures and open problems.

A note on fold thickness of graphs

A 1-fold of G is the graph G0 obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor and reducing the resulting multiple edges to simple edges. A uniform k-folding of a graph G is a sequence of graphs G = G0, G1, G2,...,Gk, where Gi+1 is a 1-fold of Gi for i = 0, 1, 2,...,k − 1 such that all graphs in the sequence are singular or all of them are nonsingular. The largest k for which there exists a uniform k- folding of G is called fold thickness of G and this concept was first introduced in [1]. In this paper, we determine fold thickness of corona product graph G ʘ Km , G ʘ S Km , and graph join G + Km .

A Method for Computing the Edge-Hosoya Polynomial with Application to Phenylenes

The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance k ≥ 0 in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph G which is obtained by identifying two edges of connected bipartite graphs G1 and G2. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.

Partitions and well-coveredness: The graph sandwich problem

A graph G is well-covered if every maximal independent set of G is maximum. A (k,ℓ)-partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k,ℓ)-well-covered if it is well-covered and admits a (k,ℓ)-partition. The recognition of (k,ℓ)-well-covered graphs is polynomial-time solvable for the cases (0,1), (0,2), (1,0), (1,1), (1,2), and (2,0), and hard, otherwise. In the graph sandwich problem for a property Π, we are given a pair of graphs G1=(V,E1) and G2=(V,E2) with E1⊆E2, and asked whether there is a graph G=(V,E) with E1⊆E⊆E2, such that G satisfies the property Π. The problem of recognizing whether a graph G satisfies a property Π is equivalent to the particular graph sandwich problem where E1=E2. In this paper, we study the graph sandwich problem for the property of being (k,ℓ)-well-covered. We present some structural characterizations and extending previous studies on the recognition of (k,ℓ)-well-covered graphs, we prove that Graph Sandwich for(k,ℓ)-well-coveredness is polynomial-time solvable when (k,ℓ)∈{(0,1),(1,0),(1,1),(0,2)}. Besides, we show that it is NP-complete for the property of being (1,2)-well-covered.

Some results on the multipartite Ramsey numbers mj(C3,Cm,n1K2,n2K2,…,niK2)

The graph Kj×t is a graph which is complete and multipartite which includes j partite sets and t vertices in each partite set. The multipartite Ramsey number (M-R-number) mj(G1,G2,…,Gn) is the smallest integer t for the mentioned graphs G1,G2,…,Gn, in a way which for each n-edge-coloring (G1,G2,…,Gn) of the edges of Kj×t, Gi contains a monochromatic copy of Gi for at least one i. The size of M-R-number mj(nK2,C7) for j≥2, n≤6, the M-R-number mj(nK2,C7) for j=2,3,4, n≥2, the M-R-number mj(nK2,C7) for each j≥5, n≥2, the M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) for j≤6, and i,ni≥1, and the size of M-R-number mj(C3,C3,nK2) for j≥2 and n≥1 have been calculated in various articles hitherto. We acquire some bounds of M-R-number mj(C3,C3,n1K2,n2K2,…,niK2) in this essay in which i,j≥2, and ni≥1, also the size of M-R-number m4(C3,C4,nK2) for each n≥1 is computed in this paper.

The g-Extra Connectivity of the Strong Product of Paths and Cycles

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph G is called a g-extra cut if G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G1=(V1,E1) and G2=(V2,E2), the strong product G1⊠G2 is defined as follows: its vertex set is V1×V2 and its edge set is {(x1,x2)(y1,y2)|x1=x2 and y1y2∈E2; or y1=y2 and x1x2∈E1; or x1x2∈E1 and y1y2∈E2}, where (x1,x2),(y1,y2)∈V1×V2. In this paper, we obtain the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.

Hom-Complex-Based Machine Learning (HCML) for the Prediction of Protein-Protein Binding Affinity Changes upon Mutation.

Protein-protein interactions (PPIs) are involved in almost all biological processes in the cell. Understanding protein-protein interactions holds the key for the understanding of biological functions, diseases and the development of therapeutics. Recently, artificial intelligence (AI) models have demonstrated great power in PPIs. However, a key issue for all AI-based PPI models is efficient molecular representations and featurization. Here, we propose Hom-complex-based PPI representation, and Hom-complex-based machine learning models for the prediction of PPI binding affinity changes upon mutation, for the first time. In our model, various Hom complexes Hom(G1, G) can be generated for the graph representation G of protein-protein complex by using different graphs G1, which reveal G1-related inner connections within the graph representation G of protein-protein complex. Further, for a specific graph G1, a series of nested Hom complexes are generated to give a multiscale characterization of the PPIs. Its persistent homology and persistent Euler characteristic are used as molecular descriptors and further combined with the machine learning model, in particular, gradient boosting tree (GBT). We systematically test our model on the two most-commonly used data sets, that is, SKEMPI and AB-Bind. It has been found that our model outperforms all the existing models as far as we know, which demonstrates the great potential of our model for the analysis of PPIs. Our model can be used for the analysis and design of efficient antibodies for SARS-CoV-2.

РАЗРАБОТКА МОДИФИЦИРАВАННЫХ МЕТОДОВ И МОДЕЛЕЙ ПОИСКОВОЙ АДАПТАЦИИ ДЛЯ РЕШЕНИЯ ЗАДАЧИ ПЛАНИРОВАНИЯ СБИС

In this work, to solve the VLSI planning problem, a search algorithm has been developedbased on a modified ant colony method. The task of forming a VLSI plan is reduced to the task offorming the corresponding Polish expression. The developed method for the synthesis of the Polishexpression includes the construction of a tree of cuts, the choice of the types of cuts (H or V), identificationand orientation of modules. The evolving population is split into pairs of agents. Eachmember of the population is a pair of agents working together. In this case, the constructive algorithmsA1 and A2 used by the agents of the pair are different. The problem solved by Algorithm A1is formulated as the problem of finding a one-to-one mapping Fk=M*→P of the set of modules Mwith selected orientations, |M*|=|M| to the set P of positions of the template Sh. In fact, the solutionconsists in choosing on the graph G1 a subset of edges E*1E1 included in the correspondingmapping Fk. In Algorithm A2, the graph G2=(X, E2) is developed as a model of the search spacefor solutions for choosing the type, sequence and location of cuts in the pattern Sh.X={(x1i,x2i)|i=1,2,…,n} the set of vertices of the graph G2, corresponds to the set P of potentialpositions of the template Sh for the possible placement of the names of the cut symbols in them.Each potential position piP of the template Sh is modeled by two alternative vertices (x1i,x2i).The choice of the vertex x1i when placing the cuts indicates that a cut of type V is placed in positionpi, the choice of vertex x2i indicates that a cut of type H is placed in position pi. Each iterationl of the general algorithm includes an initial and three main stages. The initial stage is as follows.Co-evolutionary memory matrices are nullified CEM*1 and CEM*2 are reset to zero. At the firststage, each pair of agents dk=(a1k,a2k): – with constructive algorithms A1 and A2 he synthesizeshis solution Wk=(E1k*,Sk); – the Polish expression Shk is formed, corresponding to the solution Wk;– on the basis of Shk, a tree of sections Tk is formed; – on the basis of Tk, the plan Rk is formed andthe estimate of the solution Fk is calculated; – agents deposit (add) the pheromone to the cells ofthe collective evolutionary memory (CEM) matrices CEM*1 and CEM*2 corresponding to thesolution edges Wk=(E1k*,Sk) in the solution search graphs G1 and G2 in an amount proportionalto the solution estimate Fk. At the second stage, the pheromone accumulated in CEM*1 andCEM*2 by agents of the population at iteration l is added to CEM 1 and CEM2. At the third stage,the pheromone is evaporated on the edges of the graphs G1 and G2. Tests have confirmed theeffectiveness of the proposed method. The time complexity of the algorithm, obtained experimentally,coincides with theoretical studies and it is O(n2) for the considered test problems.

A polynomial‐time algorithm for simple undirected graph isomorphism

A polynomial‐time algorithm for simple undirected graph isomorphism

Resistance distance and Kirchhoff index of the Q-vertex (or edge) join graphs

Graph operations have been extensively applied to the analysis of complex networks with properties abstracted from the real world. The resistance distance is a distance function which is related to the hitting time and random walks on graphs. In this paper, we focus on the resistance distance of a Q-vertex (or edge) join operation G1〈v〉G2 (or G1〈e〉G2) of graphs G1 and G2. When G1 is a regular graph, we give the resistance distance of the Q-vertex (or edge) join graph in terms of the number of walks in G1〈v〉G2 (or G1〈e〉G2).

ON DIRECT SUM OF FOUR FUZZY GRAPHS

The graph G1, G2 G3 and G4 which is defined by the direct sum of four fuzzy graphs G1⊕G2⊕G3⊕G4 Tins is also proved the effective values. The degree of the vertices is G1⊕G2⊕G3⊕G4 is calculated with the establishment of the regular property and the Connectedness of the Direct Sum of four Fuzzy Graphs.