Maximum Vertex Research Articles

P-Permutation equivalences between blocks of group algebras

We extend the notion of a p-permutation equivalence between two p-blocks A and B of finite groups G and H, from the definition in [5] to a virtual p-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of A and B are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that p-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of p-permutation equivalences between A and B is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of G×H arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of p-permutation bimodules.

Sharp lower bounds for the Laplacian Estrada index of graphs

Let G be a simple graph on n vertices, and let λ 1 , λ 2 , … , λ n be the Laplacian eigenvalues of G. The Laplacian Estrada index of G is defined as LEE ( G ) = ∑ i = 1 n e λ i . Consider a graph G with n ≥ 3 vertices, m edges, c connected components, and the largest Laplacian eigenvalue λ n . Let K n , S n , and K p , q (p + q = n) denote the complete graph, the star graph, and the complete bipartite graph on n vertices, respectively. In this paper, we establish that LEE ( G ) ≥ n e 2 m n + c + e λ n − ( c + 1 ) e λ n c + 1 . Furthermore, we show that the equality holds if and only if G ≅ K ¯ n (the complement of K n ), G ≅ ∪ i = 1 c − 1 K 1 ∪ S c + 1 if n = 2c, or G ≅ K n 2 , n 2 if G is a connected graph on an even number of vertices. As a consequence of this lower bound, we derive sharp lower bounds for the Laplacian Estrada index of a graph, considering its well-known graph parameters. This leads to improvements to some previously known lower bounds for the Laplacian Estrada index of a graph. Notably, we establish a sharp lower bound for the Laplacian Estrada index of a graph in terms of its maximum vertex degree. As an application, we demonstrate that the lower bound for the Laplacian Estrada index presented by Khosravanirad in [A Lower Bound for Laplacian Estrada Index of a Graph, MATCH Commun Math Comput Chem. 2013;70:175–180.] is not complete. Consequently, we provide a complete version of this lower bound.

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.

The ability of different compositions of calcium silicate and epoxy sealers to withstand gutta percha removal via in vitro pull-out testing

Objectiveexamination of the influence of chemical composition changes on the ability of sealers to withstand a pull-out test.Materials and methodsFifty distal or palatal canals of extracted teeth were prepared by Dc Taper files. The teeth were divided into five groups: AH Plus, BJM RCS, Total Fill BC,AH Plus Bioceramic and a group with Gutta Percha with no sealer added. Ten days after obturation, each cone was subjected to the “pull-out test” with the Shimadzo Universal Testing Machine until it was torn or removed from the canal. A force to Stroke graph was generated and the maximum vertex of this graph was recorded. The number of times the cone was torn or removed was recorded.ResultsThe amount of force needed to remove or rupture the cone was significantly higher in all sealer groups compared to the AH Plus Bioceramic group. The force needed for the AH Plus group was double that needed for the AH Plus Bioceramic group 4 (1.87 ± 0.53 N vs 0.93 ± 0.48 N, respectively, P < 0.001). All of the cones (n = 10) in the AH Plus Bio Ceramic Sealer group were removed in their entirety (P = 0.01 compared to each of the other groups).ConclusionsThe addition of macromolecules to epoxy sealer does not change the material’s ability to withstand the pull-out test. Decreasing the amount of tri- and di-calcium silicate compounds combined with increasing amounts of zirconium oxide in a Bioceramic sealer significantly decreased the material’s ability to withstand the pull-out test.

Seismic behavior of steel frames with H-shaped damper

This paper investigates the effect of dampers on the seismic performance of steel frames with unequal depth beam-column (UDBC) joints. First of all, the finite element (FE) models of the UDBC joint and steel frame were regenerated by ABAQUS and verified by experimental results. Secondly, the design method for three H-shaped dampers with different web perforation forms (vertical, elliptical, and diamond) was provided. Finally, five steel frames were built using ABAQUS, and pushover analysis and dynamic time-history analysis were performed on them. The results indicated that dampers could decrease the maximum vertex displacement and inter-story shift angle of steel frames with UDBC joints under three seismic waves by 4.47% to 17.17% and 7.71% to 23.88%. The effect of H-shaped dampers was similar to that of T-shaped connectors. Under the action of the Kobe wave and the Qian'an wave, the H-shaped damper exhibited more excellent seismic performance. Compared with frames with T-shaped connectors, the maximum vertex displacement and the maximum inter-story shift angle of frames with H-shaped dampers decreased by 3.44% to 12.39% and 9.55% to 16.15%. The web perforation form of H-shaped dampers somewhat affected the seismic behavior of steel frame structures. Among them, diamond dampers could minimize structural deformation.

On k-super graceful graphs with extremal maximum vertex degree

Let (,)GVE be a simple and loopless graph with ||V vertices and ||E edges. For 1,k≥ a graph G is called k-super graceful if there is a bijective labeling :fVE∪→{|ikik≤≤+||V+||E−1} with ()fvu=|()()|fvfu− for every edge .vuE∈ In this paper, we study the existence of k-super graceful labeling of certain graphs with extremal maximum vertex degree.

Study on seismic performance of nanometakaolin-reinforced fluororubber sector damper (NFSD)

In order to improve the mechanical performance of the core energy-dissipating material of the viscoelastic damper, a novel Nanometakaolin-reinforced Fluororubber Sector Damper (NFSD) is proposed for providing a strong guarantee for the seismic safety of the structure joints subjected to seismic earthquake. Firstly, in order to systematically study the shock absorption performance of NFSD, the core energy dissipation material(nano metakaolin reinforced fluororubber composite) of NFSD is prepared by melt blending method, and the vulcanization performance and microscopic mechanism of the nano composite are analyzed by vulcanization test and X-ray diffraction test. Secondly, the nanocomposite material is used as the core energy dissipation material, and NFSD based on nano reinforced fluororubber is proposed. Finally, the dynamic characteristics of NFSD and the structural seismic performance of NFSD controlled reinforced concrete (RC) frames are studied using refined finite element simulation methods. The research results indicate that 20 phr NMK increase the maximum torque MH of FKM by 20 %, while also shortening the scorching time t10 by 0.28 min. It means greatly improving the vulcanization performance of FKM. Compared to uncontrolled structures, NFSD reduces the peak displacement, interstory displacement angle, and maximum vertex acceleration of the 4-story RC structure by 16.2 %−67.4 %, 12.4 %−64.6 %, and 16.8 %−43 %, respectively. NFSD possesses excellent seismic absorption performance.

Overfullness of edge‐critical graphs with small minimal core degree

AbstractLet be a simple graph. Let and be the maximum degree and the chromatic index of , respectively. We call overfull if , and critical if for every proper subgraph of . Clearly, if is overfull then . The core of , denoted by , is the subgraph of induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer , if is critical with and , then is overfull.

On minimum spanning trees for random Euclidean bipartite graphs

AbstractWe consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$ , we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

Lower bounds on the general first Zagreb index of graphs with low cyclomatic number

The general first Zagreb index of a graph G, denoted by M1p(G), is defined as the sum of powers dGp(u) over all vertices u of V(G), where dG(u) denotes the degree of a vertex u in G. In this paper, we consider negative values of p and obtain sharp lower bounds on the general first Zagreb index of trees, unicyclic and bicyclic graphs in terms of their order and maximum vertex degrees. Also, the corresponding extremal graphs attaining the bounds are characterized. The results are then extended to other indices defined as sums over all vertices of contributions which are convex or concave functions of their degrees.

A heuristic algorithm using tree decompositions for the maximum happy vertices problem

A heuristic algorithm using tree decompositions for the maximum happy vertices problem

(n,m)-Graphs with Maximum Vertex–Degree Function–Index for Convex Functions

(n,m)-Graphs with Maximum Vertex–Degree Function–Index for Convex Functions

A new parallel tabu search algorithm for the optimization of the maximum vertex weight clique problem

SummaryThe efficiency of metaheuristic algorithms depends significantly on the number of fitness value evaluations performed on candidate solutions. In addition to various intelligent techniques used to obtain better results, parallelization of calculations can substantially improve the solutions in cases where the problem is NP‐hard and requires many evaluations. This study proposes a new parallel tabu search method for solving the Maximum Vertex Weight Clique Problem (MVWCP) on the Non‐Uniform Memory Access (NUMA) architectures using the OpenMP parallel programming paradigm. Achieving scalability in the NUMA architectures presents significant challenges due to the high complexity of their memory systems, which can lead to performance loss. However, our proposed Tabu‐NUMA algorithm provides up to speed‐up with 64 cores for ten basic problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. And it improves the performance of the serial Multi Neighborhood Tabu Search (MN/TS) algorithm for 38 problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. We further evaluate our algorithm on larger datasets with thousands of edges and vertices from Network Data Repository benchmark problem instances, and we report significant improvements in terms of speed up. Our results confirm that the Tabu‐NUMA algorithm is among the best recent algorithms for solving MVWCP on the NUMA architectures.

Relation between the trace norm of an oriented graph and its rank

Let D be an oriented graph, i.e., a digraph in which all arcs are not symmetric, with vertex set {1,…,n}. The adjacency matrix A=(aij) of D is the n×n matrix defined as aij=1 if ij is an arc of D and aij=0 if ij is not an arc of D. The rank of D, written as r(D), is defined to be the rank of its adjacency matrix, and the trace norm of D is defined as N(D)=∑i=1nσi, where σ1≥σ2≥…≥σn≥0 are the singular values of A, i.e., the square roots of the eigenvalues of AAT. In this paper, we establish a lower bound and an upper bound for the trace norm of a connected oriented graph D in terms of its rank and its maximum vertex degree Δ asr(D)≤N(D)≤r(D)Δ, the extremal graphs attaining the bounds are characterized, respectively. Furthermore, we prove that if D is not Pn→ or Cn→, then the lower bound can be improved to r(D)+5−2, where Pn→ and Cn→ respectively denote the orientation of Pn and Cn such that all 2-degree vertices are not sink-source.

Semi-Total Point Graph of Neighbourhood Edge Corona Graph of G and H

A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. Here we concentrated on topological indices involving the number of vertices, the number of edges and the maximum and minimum vertex degree. The aim of this paper is to compute the lower and upper bounds of the second Zagreb index, third Zagreb index, Hyper Zagreb index, Harmonic index, Redefined first Zagreb index, First reformulated Zagreb index, Forgotten topological index, square F-index, Sum-connectivity index, Randic index, Reciprocal Randic index, Gourava index, Sombar index, Nirmala index, Geometric-Arithmetic index and lower bonds of Atom bond connectivity index, Redefined second Zagreb.

A nonvanishing spectral gap for AKLT models on generalized decorated graphs

We consider the spectral gap question for Affleck, Kennedy, Lieb, and Tasaki models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of G with a chain of n sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States approach from [Abdul-Rahman et al., Analytic Trends in Mathematical Physics, Contemporary Mathematics (American Mathematical Society, 2020), Vol. 741, p. 1.], we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.

Investigation on Seismic Performance of the Fully Assembled Steel Frame Applying Beam-Column Joints with Replaceable Energy-Dissipating Elements

Based on the application of a beam-column joint with replaceable energy-dissipating elements and hinged beam-column proposed by the author, the seismic performance of a fully assembled steel frame with this joint was investigated in this paper. Through two projects of a traditional steel frame (TSF) and an assembled steel frame applying beam-column joints with replaceable energy-dissipating elements (ASFWREE) and their numerical simulation calculation by SAP2000, the main structural design indicators such as natural vibration period, period ratio, mass participation coefficient, base shear force, inter-story displacement angle, rigid-weight ratio, and shear-weight ratio of two frames under frequent earthquake, and their influence factors were compared and analyzed. By carrying out the elastic–plastic dynamic time–history analysis of two projects under a rare earthquake, the maximum inter-story displacement angle, base shear force, stiffness between floors, maximum vertex displacement, and the occurrence sequence and distribution of plastic hinges of the two items were compared. The results show that the deformation of ASFWREE had the characteristics of the shear deformation of TSF and the deviation of the natural vibration period was less than 5% when the ratio of the linear stiffness of the energy-dissipating element to the steel beam was approximately 0.8, the ratio of horizontal length to span was about 0.25, which was close to the strength grade of the steel beam. The seismic performance under the rare earthquake was close to or higher to that of the TSF, can ensure that the beam and column are not damaged, and the structure does not collapse. The failure mode of the ASFWREE is consistent with the experimental research of the beam-column joints.

Research on Matching and Vertex Cover Problems in Bipartite Graphs using Simplex Method

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Minimizing the Average Packet Access Time of the Application Layer for Buffered Instantly Decodable Network Coding

In B-IDNC (buffered instantly decodable network coding), each receiver can cache the non-instantly decodable network coded packets (NIDPs) which contain two wanted packets of the network layer for subsequent network decoding, so that more packets can be recovered at the network layer than the traditional instantly decodable network coding (IDNC). By employing B-IDNC, we consider a radio access network wherein a base station (BS) is required to broadcast a block of packets to a set of receivers. After completing network decoding at the network layer, each receiver can deliver its recovered packets from the network layer to the application layer in order. We consider minimizing the average packet access time of the application layer for B-IDNC. For the optimization problem is intractable, we approximate it to reduce the sum minimum access delay of the application layer across all receivers. The approximate problem is shown to be equivalent to a maximum weight encoding clique problem over the B-IDNC graph. We propose a simple heuristic algorithm based on greedy maximum weight vertex search to solve the approximate problem. Simulation results verify the effectiveness of our proposed algorithm as compared with the existing techniques.

On solving simplified diversified top-[formula omitted][formula omitted]-plex problem

Finding cohesive groups in a graph, which has been extensively studied by many researchers, is a fundamental and critical problem for various real-world applications, such as community search, motif discovery and anomaly detection. Unfortunately, the cohesive groups rarely appear as cliques and are usually highly overlapping, so few cohesive groups can be found by searching maximum or top-k maximal cliques. In other words, maximum or top-k maximal cliques are too strict for representing cohesive groups. To handle this problem, the DTKSP problem was introduced earlier in the literature to find k maximal s-plexes that cover maximum vertices with the lowest overlapping in a given graph. In this paper, we consider the Simplified Diversified Top-ks-Plex (S-DTKSP) problem, by aiming to find k maximal s-plexes that cover the maximum vertices without considering the size of overlap. We prove that the S-DTKSP problem is NP-hard and propose an integer linear programming for S-DTKSP problem. Then, we propose an iterated local search (ILS) algorithm with a tabu strategy to efficiently find a good solution. The proposed algorithms are evaluated on large real-world instances. The experimental results demonstrate that our approaches can solve the S-DTKSP problem effectively and efficiently than two baseline algorithms.