This paper presents an optimisatiombased verification process for obstacle avoidance systems of a unicycle-like mobile robot. It is a novel approach for the collision avoidance verification process. Local and global o...This paper presents an optimisatiombased verification process for obstacle avoidance systems of a unicycle-like mobile robot. It is a novel approach for the collision avoidance verification process. Local and global optimisation based verification processes are developed to find the worst-case parameters and the worst-case distance between the robot and an obstacle. The kinematic and dynamic model of the unicycle-like mobile robot is first introduced with force and torque as the inputs. The design of the control system is split into two parts. One is velocity and rotation using the robot dynamics, and the other is the incremental motion planning for robot kinematics. The artificial potential field method is chosen as a path planning and obstacle avoidance candidate technique for verification study as it is simple and widely used. Different optimisation algorithms are applied and compared for the purpose of verification. It is shown that even for a simple case study where only mass and inertia variations are considered, a local optimization based verification method may fail to identify the worst case. Two global optimisation methods have been investigated： genetic algorithms （GAs） and GLOBAL algorithms. Both of these methods successfully find the worst case. The verification process confirms that the obstacle avoidance algorithm functions correctly in the presence of all the possible parameter variations.展开更多
This paper discusses about balancing control of unicycle robot.Unicycle robot consists of pitch which acts like inverted pendulum and roll which acts like reaction wheel pendulum.The robot which does not have actuator...This paper discusses about balancing control of unicycle robot.Unicycle robot consists of pitch which acts like inverted pendulum and roll which acts like reaction wheel pendulum.The robot which does not have actuator located in yaw axis is made to derive the simple dynamics.Lagrange equation is applied to deriving dynamic equations.Obtained dynamic equations are used to design the sliding mode control.State variables of the designed control are pitch angle and roll angle.Sliding mode control has chattering problem,which is eliminated by using the sigmoid function as switching function.Finally the control performance and eliminated chattering problem is verified by simulation.展开更多
The concept of an intelligent control system for a complex nonlinear biomechanical system of an extension cableless robotic unicycle discussed.A thermodynamic approach to study optimal control processes in complex non...The concept of an intelligent control system for a complex nonlinear biomechanical system of an extension cableless robotic unicycle discussed.A thermodynamic approach to study optimal control processes in complex nonlinear dynamic systems applied.The results of stochastic simulation of a fuzzy intelligent control system for various types of external/internal excitations for a dynamic,globally unstable control object-extension cableless robotic unicycle based on Soft Computing(Computational Intelligence Toolkit-SCOptKBTM)technology presented.A new approach to design an intelligent control system based on the principle of the minimum entropy production(minimum of useful resource losses)determination in the movement of the control object and the control system is developed.This determination as a fitness function in the genetic algorithm is used to achieve robust control of a robotic unicycle.An algorithm for entropy production computing and representation of their relationship with the Lyapunov function(a measure of stochastic robust stability)described.展开更多
Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is s...Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is studied. It is proved a piecewise linear function of the order and edge size of the hypergraph.展开更多
In this paper,the formation control problem for unicycle multi-agent systems is considered.The design of a generalized homogeneous leader–follower formation control protocol is studied which shows that such a control...In this paper,the formation control problem for unicycle multi-agent systems is considered.The design of a generalized homogeneous leader–follower formation control protocol is studied which shows that such a control protocol can be obtained by“upgrading”from the classical linear control.With this proposed control protocol,a finite-time stability of the formation error is ensured if the acceleration of the leader is bounded by some known value.In addition,if the leader acceleration is unknown but bounded,the robust formation is achieved by ensuring the formation error's input-to-state stability.Simulations are carried out to verify the effectiveness of the proposed control protocol.展开更多
The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G ...The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.展开更多
Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to elim...Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to eliminate the risk of human injuries while working on hazardous tasks, and to improve productivity. Globally, engineers are continuously researching better, simple, and faster cooperative Control algorithms to provide a Control strategy where each agent in the robot formation can communicate effectively and achieve a consensus in their position, orientation and speed.<span style="font-family:Verdana;"> </span><span style="font-family:Verdana;">This paper explores a novel Formation Building Algorithm and its global stability around a configuration vector. A simulation in MATLAB</span><sup><span style="font-size:12px;font-family:Verdana;"><span lang="ZH-CN" style="font-size:12pt;font-family:宋体;">?</span></span></sup><span style="font-family:Verdana;"> was carried out to examine the performance of the Algorithm for two geometric formations and a fixed number of robots. In addition, an obstacle avoidance technique was presented assuming that all robots are equipped with range sensors. In particular, a uniform rounded obstacle is used to analyze</span><span style="font-family:Verdana;"> </span><span style="font-family:Verdana;">the performance of the technique with the use of detailed geometric calculations.</span>展开更多
The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U...The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.展开更多
The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the comple...The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.展开更多
Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its compleme...Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.展开更多
A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic the...A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.展开更多
Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index （AZI index） of a graph G, which is defined asAZI（G） = ∑uv∈E...Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index （AZI index） of a graph G, which is defined asAZI（G） = ∑uv∈E（G）（dudv/du＋du-2）3,where E（G） is the edge set of G, d~ and d~ are the degrees of the terminal vertices u and v of edge uv, respectively. In this paper, we obtain the first five largest （resp., the first two smallest） AZI indices of connected graphs with n vertices. Moreover, we determine the trees of order n with the first three smallest AZI indices, the unicyclic graphs of order n with the minimum, the second minimum AZI indices, and the bicyclic graphs of order n with the minimum AZI index, respectively.展开更多
In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wie...In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl. Math., 2014, 171：35 41）, and give a simple way to compare the Wiener index and the Zagreb indices for trees. Moreover, the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.展开更多
Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved t...Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H （n; q, n1, n2, n3, n4） are determined by their Laplacian spectra.展开更多
The Balaban index of a connected graph G is defined as J（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）DG（v）,and the Sum-Balaban index is defined as SJ（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）＋DG（v）,where...The Balaban index of a connected graph G is defined as J（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）DG（v）,and the Sum-Balaban index is defined as SJ（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）＋DG（v）,where DG（u） =∑w∈V（G）dG（u, w）, and μ is the cyclomatic number of G. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on n vertices are characterized, respectively.展开更多
For a simple graph G, the energy E（G） is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un...For a simple graph G, the energy E（G） is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un｜ d（x） = r for any vertex x ∈ V（Cl）}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39（2, 2, 2） ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.展开更多
A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH（G）,is referred to as the multiplicity of the eigenval...A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH（G）,is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH（G）, which combines the cases of undirected and oriented unicyclic graphs and also corrects an error in Theorem 4.2 of [Xueliang LI, Guihai YU. The skew-rank of oriented graphs. Sci. Sin. Math., 2015, 45：93-104（in Chinese）]. In addition, we characterize all the n-vertex mixed graphs with nullity n-3, which are determined by the spectrum of their Hermitian-adjacency matrices.展开更多
Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that the...Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V（G） such that Q（G）f（u） = λf（u） for u ∈ V_0 and f（u） = 0 for u ∈ ？V. The signless Dirichlet spectral radiusλ（G） is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.展开更多
A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic grap...A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic graph and dG(u0)=d0}.Let A(G) be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G) be the eigenvalues in non-increasing order of A(G).The number n∑i=1λi^k(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by Sk(G).Let S(G)=(S0(G),S1(G),…,Sn-1(G)) be the sequence of spectral moments of G.For two graphs G1,G2,we have■ if for some k(k=1,2,…,n-1), and we have Si(G1)=Si(G2)(i=0,1,…,k-1) and Sk(G1)<Sk(G2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d0),respectively.展开更多
Another version of atom-bond connectivity index was defined by Graovac and Ghorbani, and called the second atom-bond connectivity index(ABC_2), which can provide convenience for molecular feature and its extreme value...Another version of atom-bond connectivity index was defined by Graovac and Ghorbani, and called the second atom-bond connectivity index(ABC_2), which can provide convenience for molecular feature and its extreme values are the focus of study. In this paper, by fractional comparison, we give the structural properties of the extremal graphs which attain the minimal ABC_2 index of unicyclic graphs of order n.展开更多
文摘This paper presents an optimisatiombased verification process for obstacle avoidance systems of a unicycle-like mobile robot. It is a novel approach for the collision avoidance verification process. Local and global optimisation based verification processes are developed to find the worst-case parameters and the worst-case distance between the robot and an obstacle. The kinematic and dynamic model of the unicycle-like mobile robot is first introduced with force and torque as the inputs. The design of the control system is split into two parts. One is velocity and rotation using the robot dynamics, and the other is the incremental motion planning for robot kinematics. The artificial potential field method is chosen as a path planning and obstacle avoidance candidate technique for verification study as it is simple and widely used. Different optimisation algorithms are applied and compared for the purpose of verification. It is shown that even for a simple case study where only mass and inertia variations are considered, a local optimization based verification method may fail to identify the worst case. Two global optimisation methods have been investigated： genetic algorithms （GAs） and GLOBAL algorithms. Both of these methods successfully find the worst case. The verification process confirms that the obstacle avoidance algorithm functions correctly in the presence of all the possible parameter variations.
基金The MKE(The Ministry of Knowledge Economy),Korea,under the ITRC(Infor mation Technology Research Center)support program supervised by the NIPA(National IT Industry Promotion Agency)(NIPA-2012-H0301-12-2006)the MKE,Korea,under the Human Resources Development Programfor Specialized Environment Navigation/Localization Technology Research Center support program supervised by the NIPA(NIPA-2012-H1502-12-1002)
文摘This paper discusses about balancing control of unicycle robot.Unicycle robot consists of pitch which acts like inverted pendulum and roll which acts like reaction wheel pendulum.The robot which does not have actuator located in yaw axis is made to derive the simple dynamics.Lagrange equation is applied to deriving dynamic equations.Obtained dynamic equations are used to design the sliding mode control.State variables of the designed control are pitch angle and roll angle.Sliding mode control has chattering problem,which is eliminated by using the sigmoid function as switching function.Finally the control performance and eliminated chattering problem is verified by simulation.
文摘The concept of an intelligent control system for a complex nonlinear biomechanical system of an extension cableless robotic unicycle discussed.A thermodynamic approach to study optimal control processes in complex nonlinear dynamic systems applied.The results of stochastic simulation of a fuzzy intelligent control system for various types of external/internal excitations for a dynamic,globally unstable control object-extension cableless robotic unicycle based on Soft Computing(Computational Intelligence Toolkit-SCOptKBTM)technology presented.A new approach to design an intelligent control system based on the principle of the minimum entropy production(minimum of useful resource losses)determination in the movement of the control object and the control system is developed.This determination as a fitness function in the genetic algorithm is used to achieve robust control of a robotic unicycle.An algorithm for entropy production computing and representation of their relationship with the Lyapunov function(a measure of stochastic robust stability)described.
文摘Cyclic hypergraphs are the analogue of cyclic graphs. The unicycle hypergraph whose cyclomatic number equals to one is the most elemental cyclic hypergraph. In this paper the maximum size of unicycle hypergraphs is studied. It is proved a piecewise linear function of the order and edge size of the hypergraph.
基金supported by the CSC Grant 202106160022the SEP-CONACYT-ANUIES-ECOS NORD Project 315597+2 种基金ECOS NORD Project M20M04supported in part by the CONAHCYT CVU 270504 Project 922in part by the TecNM Projects
文摘In this paper,the formation control problem for unicycle multi-agent systems is considered.The design of a generalized homogeneous leader–follower formation control protocol is studied which shows that such a control protocol can be obtained by“upgrading”from the classical linear control.With this proposed control protocol,a finite-time stability of the formation error is ensured if the acceleration of the leader is bounded by some known value.In addition,if the leader acceleration is unknown but bounded,the robust formation is achieved by ensuring the formation error's input-to-state stability.Simulations are carried out to verify the effectiveness of the proposed control protocol.
基金Supported by the National Natural Science Foundation of China(Nos.12271439,11871398)the National College Students Innovation and Entrepreneurship Training Program(No.201910699173)。
文摘The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.
文摘Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to eliminate the risk of human injuries while working on hazardous tasks, and to improve productivity. Globally, engineers are continuously researching better, simple, and faster cooperative Control algorithms to provide a Control strategy where each agent in the robot formation can communicate effectively and achieve a consensus in their position, orientation and speed.<span style="font-family:Verdana;"> </span><span style="font-family:Verdana;">This paper explores a novel Formation Building Algorithm and its global stability around a configuration vector. A simulation in MATLAB</span><sup><span style="font-size:12px;font-family:Verdana;"><span lang="ZH-CN" style="font-size:12pt;font-family:宋体;">?</span></span></sup><span style="font-family:Verdana;"> was carried out to examine the performance of the Algorithm for two geometric formations and a fixed number of robots. In addition, an obstacle avoidance technique was presented assuming that all robots are equipped with range sensors. In particular, a uniform rounded obstacle is used to analyze</span><span style="font-family:Verdana;"> </span><span style="font-family:Verdana;">the performance of the technique with the use of detailed geometric calculations.</span>
基金Supported by the project item for young teachers of colleges and universities of Anhui province( 2 0 0 3jq1 0 1 ) and the project item of Anhui University for talents group construction
文摘The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.
基金Supported by the National Natural Science Foundation of China(11861011,11501133,11661010)。
文摘The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.
基金Supported by the National Natural Science Foundation of China(11871256)the Chinese-Croatian bilateral project(7-22)。
文摘Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.
基金Supported by the National Natural Science Foundation of China (10671081)
文摘A graph has exactly two main eigenvalues if and only if it is a 2-walk linear graph.In this paper,we show some necessary conditions that a 2-walk（a,b）-linear graph must obey.Using these conditions and some basic theorems in graph theory,we characterize all 2-walk linear graphs with small cyclic graphs without pendants.The results are given in sort on unicyclic,bicyclic,tricyclic graphs.
基金Supported by the National Natural Science Foundation of China(Grant No.11326221)
文摘Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index （AZI index） of a graph G, which is defined asAZI（G） = ∑uv∈E（G）（dudv/du＋du-2）3,where E（G） is the edge set of G, d~ and d~ are the degrees of the terminal vertices u and v of edge uv, respectively. In this paper, we obtain the first five largest （resp., the first two smallest） AZI indices of connected graphs with n vertices. Moreover, we determine the trees of order n with the first three smallest AZI indices, the unicyclic graphs of order n with the minimum, the second minimum AZI indices, and the bicyclic graphs of order n with the minimum AZI index, respectively.
基金The NSF(11301093,11501139)of Chinathe NSF(2014A030313640)of Guangdong Provincethe Foundation(Yq2014111)for Distinguished Young Talents in Higher Education of Guangdong Province
文摘In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl. Math., 2014, 171：35 41）, and give a simple way to compare the Wiener index and the Zagreb indices for trees. Moreover, the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.
基金the National Natural Science Foundation of China(Grant No.11171273)Graduate StartingSeed Fund of Northwestern Polytechnical University(Grant No.Z2014173)
文摘Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H （n; q, n1, n2, n3, n4） are determined by their Laplacian spectra.
基金Supported by the Zhujiang Technology New Star Foundation of Guangzhou(Grant No.2011J2200090)Program on International Cooperation and Innovation,Department of Education,Guangdong Province(GrantNo.2012gjhz0007)
文摘The Balaban index of a connected graph G is defined as J（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）DG（v）,and the Sum-Balaban index is defined as SJ（G） =｜E（G）｜μ ＋ 1∑e=uv∈E（G）1√DG（u）＋DG（v）,where DG（u） =∑w∈V（G）dG（u, w）, and μ is the cyclomatic number of G. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on n vertices are characterized, respectively.
基金Supported by the National Natural Science Foundation of China(Grant No.11326216)the Docter Foundationof Shandong University of Technology(Grant No.413010)
文摘For a simple graph G, the energy E（G） is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un｜ d（x） = r for any vertex x ∈ V（Cl）}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39（2, 2, 2） ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.
基金Supported by the National Natural Science Foundation of China(Grant No.11571360)
文摘A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH（G）,is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH（G）, which combines the cases of undirected and oriented unicyclic graphs and also corrects an error in Theorem 4.2 of [Xueliang LI, Guihai YU. The skew-rank of oriented graphs. Sci. Sin. Math., 2015, 45：93-104（in Chinese）]. In addition, we characterize all the n-vertex mixed graphs with nullity n-3, which are determined by the spectrum of their Hermitian-adjacency matrices.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1127125611601208)
文摘Let G be a simple connected graph with pendant vertex set ？V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q（G）. The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V（G） such that Q（G）f（u） = λf（u） for u ∈ V_0 and f（u） = 0 for u ∈ ？V. The signless Dirichlet spectral radiusλ（G） is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.
基金Supported by China Scholarship Council(201808420093)Research Project of Education Bureau of Wuhan(2014017)
文摘A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic graph and dG(u0)=d0}.Let A(G) be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G) be the eigenvalues in non-increasing order of A(G).The number n∑i=1λi^k(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by Sk(G).Let S(G)=(S0(G),S1(G),…,Sn-1(G)) be the sequence of spectral moments of G.For two graphs G1,G2,we have■ if for some k(k=1,2,…,n-1), and we have Si(G1)=Si(G2)(i=0,1,…,k-1) and Sk(G1)<Sk(G2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d0),respectively.
文摘Another version of atom-bond connectivity index was defined by Graovac and Ghorbani, and called the second atom-bond connectivity index(ABC_2), which can provide convenience for molecular feature and its extreme values are the focus of study. In this paper, by fractional comparison, we give the structural properties of the extremal graphs which attain the minimal ABC_2 index of unicyclic graphs of order n.
