• Previous Article
    A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems
  • NACO Home
  • This Issue
  • Next Article
    Local smooth representation of solution sets in parametric linear fractional programming problems
March  2019, 9(1): 53-69. doi: 10.3934/naco.2019005

On construction of upper and lower bounds for the HOMO-LUMO spectral gap

1. 

Institute of Information Engineering, Automation, and Mathematics, FCFT, Slovak Technical University, 812 37 Bratislava, Slovakia

2. 

Department of Applied Mathematics and Statistics, FMFI, Comenius University, 842 48 Bratislava, Slovakia

* Corresponding author: Daniel Ševčovič

Received  January 2018 Revised  June 2018 Published  October 2018

Fund Project: The authors were supported by VEGA grants 1/0142/17(S.P.) and 1/0062/18(D.Š.).

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.

Citation: Soña Pavlíková, Daniel Ševčovič. On construction of upper and lower bounds for the HOMO-LUMO spectral gap. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 53-69. doi: 10.3934/naco.2019005
References:
[1]

J. I. Aihara, Reduced HOMO-LUMO Gap as an Index of Kinetic Stability for Polycyclic Aromatic Hydrocarbons, J. Phys. Chem. A, 103 (1999), 7487-7495.   Google Scholar

[2]

J. I. Aihara, Weighted HOMO-LUMO energy separation as an index of kinetic stability for fullerenes, Theor. Chem. Acta, 102 (1999), 134-138.   Google Scholar

[3]

N. C. Bacalis and A. D. Zdetsis, Properties of hydrogen terminated silicon nanocrystals via a transferable tight-binding Hamiltonian, based on ab-initio results, J. Math. Chem., 26 (2009), 962-970.  doi: 10.1007/s10910-009-9557-x.  Google Scholar

[4]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press New York, NY, USA, 2004. doi: 10.1017/CBO9780511804441.  Google Scholar

[5]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer New York, Dordrecht, Heidelberg, London, 2012. doi: 10.1007/978-1-4614-1939-6.  Google Scholar

[6]

D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs - Theory and Application, Academic Press, New York, 1980.  Google Scholar

[7]

D. CvetkovićP. Hansen and V. Kovačevič-Vučič, On some interconnections between combinatorial optimization and extremal graph theory, Yugoslav Journal of Operations Research, 14 (2004), 147-154.  doi: 10.2298/YJOR0402147C.  Google Scholar

[8]

P. W. FowlerP. HansenG. Caporosi and A. Soncini, Polyenes with maximum HOMO-LUMO gap, Chemical Physics Letters, 342 (2001), 105-112.   Google Scholar

[9]

P. V. Fowler, HOMO-LUMO maps for chemical graphs, MATCH Commun. Math. Comput. Chem., 64 (2010), 373-390.   Google Scholar

[10]

C. D. Godsil, Inverses of trees, Combinatorica, 5 (1985), 33-39.  doi: 10.1007/BF02579440.  Google Scholar

[11]

I. Gutman and D. H. Rouvray, An Aproximate TopologicaI Formula for the HOMO-LUMO Separation in Alternant Hydrocarboons, Chemical-Physic Letters, 72 (1979), 384-388.   Google Scholar

[12]

E. Hückel, Quantentheoretische Beiträge zum Benzolproblem, Zeitschrift für Physik, 30 (1931), 204-286.   Google Scholar

[13]

G. Jaklić, HL-index of a graph, Ars Mathematica Contemporanea, 5 (2012), 99-105.   Google Scholar

[14]

S. KimM. Kojima and K. Toh, A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems, Mathematical Programming, 156 (2016), 161-187.  doi: 10.1007/s10107-015-0874-5.  Google Scholar

[15]

Xueliang LiYiyang LiYongtang Shi and I. Gutman, Note on the HOMO-LUMO index of graphs, MATCH Commun. Math. Comput. Chem., 70 (2013), 85-96.   Google Scholar

[16]

Chen Lin and Jinfeng Liu, Extremal values of matching energies of one class of graphs, Applied Mathematics and Computation, 273 (2016), 976-992.  doi: 10.1016/j.amc.2015.10.025.  Google Scholar

[17]

L. Löfberg, A toolbox for modeling and optimization in MATLAB, 2004 IEEE internationalsymposium on computer aided control systems design (CACSD 2004), September 2-4, 2004, Taipei, 284-289. Google Scholar

[18]

M. Hamala and M. Trnovská, Nonlinear Programming, Theory and Algorithms (in Slovak), Epos, Bratislava, 2013. Google Scholar

[19]

B. Mohar, Median eigenvalues of bipartite planar graphs, MATCH Commun. Math. Comput. Chem., 70 (2013), 79-84.   Google Scholar

[20]

M. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory, Series B, 112 (2015), 78-92.  doi: 10.1016/j.jctb.2014.12.001.  Google Scholar

[21]

S. Pavlíková and J. Krč-Jediný, On the inverse and dual index of a tree, Linear and Multilinear Algebra, 28 (1990), 93-109.  doi: 10.1080/03081089008818034.  Google Scholar

[22]

S. Pavlíková, A note on inverses of labeled graphs, Australasian Journal on Combinatorics, 67 (2017), 222-234.   Google Scholar

[23]

S. Pavlíková and D. Ševčovič, On a construction of integrally invertible graphs and their spectral properties, Linear Algebra and its Applications, 532 (2017), 512-533.  doi: 10.1016/j.laa.2017.07.005.  Google Scholar

[24]

S. Pavlíková and D. Ševčovič, Maximization of the spectral gap for chemical graphs by means of a solution to a mixed integer semidefinite program, Computer Methods in Materials Science, 4 (2016), 169-176.   Google Scholar

[25]

D. Ševčovič and M. Trnovská, Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method, Journal of Inverse and Ⅲ-posed Problems, 23 (2015), 263-285.  doi: 10.1515/jiip-2013-0069.  Google Scholar

[26]

J. F. Sturm, Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones, Optimization Methods and Software, 11 (1999), 625-653.  doi: 10.1080/10556789908805766.  Google Scholar

[27]

F. Zhang and Z. Chen, Ordering graphs with small index and its application, Discrete Applied Mathematics, 121 (2002), 295-306.  doi: 10.1016/S0166-218X(01)00302-X.  Google Scholar

[28]

F. Zhang and C. An, Acyclic molecules with greatest HOMOLUMO separation, Discrete Applied Mathematics, 98 (1999), 165-171.  doi: 10.1016/S0166-218X(99)00119-5.  Google Scholar

show all references

References:
[1]

J. I. Aihara, Reduced HOMO-LUMO Gap as an Index of Kinetic Stability for Polycyclic Aromatic Hydrocarbons, J. Phys. Chem. A, 103 (1999), 7487-7495.   Google Scholar

[2]

J. I. Aihara, Weighted HOMO-LUMO energy separation as an index of kinetic stability for fullerenes, Theor. Chem. Acta, 102 (1999), 134-138.   Google Scholar

[3]

N. C. Bacalis and A. D. Zdetsis, Properties of hydrogen terminated silicon nanocrystals via a transferable tight-binding Hamiltonian, based on ab-initio results, J. Math. Chem., 26 (2009), 962-970.  doi: 10.1007/s10910-009-9557-x.  Google Scholar

[4]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press New York, NY, USA, 2004. doi: 10.1017/CBO9780511804441.  Google Scholar

[5]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer New York, Dordrecht, Heidelberg, London, 2012. doi: 10.1007/978-1-4614-1939-6.  Google Scholar

[6]

D. Cvetković, M. Doob and H. Sachs, Spectra of Graphs - Theory and Application, Academic Press, New York, 1980.  Google Scholar

[7]

D. CvetkovićP. Hansen and V. Kovačevič-Vučič, On some interconnections between combinatorial optimization and extremal graph theory, Yugoslav Journal of Operations Research, 14 (2004), 147-154.  doi: 10.2298/YJOR0402147C.  Google Scholar

[8]

P. W. FowlerP. HansenG. Caporosi and A. Soncini, Polyenes with maximum HOMO-LUMO gap, Chemical Physics Letters, 342 (2001), 105-112.   Google Scholar

[9]

P. V. Fowler, HOMO-LUMO maps for chemical graphs, MATCH Commun. Math. Comput. Chem., 64 (2010), 373-390.   Google Scholar

[10]

C. D. Godsil, Inverses of trees, Combinatorica, 5 (1985), 33-39.  doi: 10.1007/BF02579440.  Google Scholar

[11]

I. Gutman and D. H. Rouvray, An Aproximate TopologicaI Formula for the HOMO-LUMO Separation in Alternant Hydrocarboons, Chemical-Physic Letters, 72 (1979), 384-388.   Google Scholar

[12]

E. Hückel, Quantentheoretische Beiträge zum Benzolproblem, Zeitschrift für Physik, 30 (1931), 204-286.   Google Scholar

[13]

G. Jaklić, HL-index of a graph, Ars Mathematica Contemporanea, 5 (2012), 99-105.   Google Scholar

[14]

S. KimM. Kojima and K. Toh, A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems, Mathematical Programming, 156 (2016), 161-187.  doi: 10.1007/s10107-015-0874-5.  Google Scholar

[15]

Xueliang LiYiyang LiYongtang Shi and I. Gutman, Note on the HOMO-LUMO index of graphs, MATCH Commun. Math. Comput. Chem., 70 (2013), 85-96.   Google Scholar

[16]

Chen Lin and Jinfeng Liu, Extremal values of matching energies of one class of graphs, Applied Mathematics and Computation, 273 (2016), 976-992.  doi: 10.1016/j.amc.2015.10.025.  Google Scholar

[17]

L. Löfberg, A toolbox for modeling and optimization in MATLAB, 2004 IEEE internationalsymposium on computer aided control systems design (CACSD 2004), September 2-4, 2004, Taipei, 284-289. Google Scholar

[18]

M. Hamala and M. Trnovská, Nonlinear Programming, Theory and Algorithms (in Slovak), Epos, Bratislava, 2013. Google Scholar

[19]

B. Mohar, Median eigenvalues of bipartite planar graphs, MATCH Commun. Math. Comput. Chem., 70 (2013), 79-84.   Google Scholar

[20]

M. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, Journal of Combinatorial Theory, Series B, 112 (2015), 78-92.  doi: 10.1016/j.jctb.2014.12.001.  Google Scholar

[21]

S. Pavlíková and J. Krč-Jediný, On the inverse and dual index of a tree, Linear and Multilinear Algebra, 28 (1990), 93-109.  doi: 10.1080/03081089008818034.  Google Scholar

[22]

S. Pavlíková, A note on inverses of labeled graphs, Australasian Journal on Combinatorics, 67 (2017), 222-234.   Google Scholar

[23]

S. Pavlíková and D. Ševčovič, On a construction of integrally invertible graphs and their spectral properties, Linear Algebra and its Applications, 532 (2017), 512-533.  doi: 10.1016/j.laa.2017.07.005.  Google Scholar

[24]

S. Pavlíková and D. Ševčovič, Maximization of the spectral gap for chemical graphs by means of a solution to a mixed integer semidefinite program, Computer Methods in Materials Science, 4 (2016), 169-176.   Google Scholar

[25]

D. Ševčovič and M. Trnovská, Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method, Journal of Inverse and Ⅲ-posed Problems, 23 (2015), 263-285.  doi: 10.1515/jiip-2013-0069.  Google Scholar

[26]

J. F. Sturm, Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones, Optimization Methods and Software, 11 (1999), 625-653.  doi: 10.1080/10556789908805766.  Google Scholar

[27]

F. Zhang and Z. Chen, Ordering graphs with small index and its application, Discrete Applied Mathematics, 121 (2002), 295-306.  doi: 10.1016/S0166-218X(01)00302-X.  Google Scholar

[28]

F. Zhang and C. An, Acyclic molecules with greatest HOMOLUMO separation, Discrete Applied Mathematics, 98 (1999), 165-171.  doi: 10.1016/S0166-218X(99)00119-5.  Google Scholar

Figure 1.  A bridged graph $G_C = {\mathcal B}_K(G_A, G_B)$ through a bipartite graph $G_K$.
Figure 2.  Simple graphs $G_A$ and $G_B$ (left) and the bridged graph $G_C$ with the maximal HOMO-LUMO spectral gap which can be constructed by bridging $G_A$ and $G_B$ over the vertex 1 of $G_B$ ($k_B = 1$) to the vertices of $G_A$ (right).
Figure 3.  An example of an invertible graph $F_0$ (left) representing the chemical organic molecule of fulvene (right).
Figure 4.  Results of optimal bridging of the fulvene graph GB= F0 through the vertices {1,2} to GA=F0 a);through the vertices {1,4} to GA=F0 b);and through the vertices {1,2} to GA=F1 c).
Figure 5.  Results of optimal bridging of the graph $G_B = P_4$ through the vertices $\{1, 3\}$ to $G_A = P_6$ a); through the vertices $\{2, 3\}$ to $G_A = P_6$ b); and through the vertex $\{2\}$ to $G_A = T_4$ c).
Figure 6.  Results of optimal bridging of the graph $G_B = P_4$ through the vertices $\{1, 3\}$ to $G_A = P_6$ a); through the vertices $\{2, 3\}$ to $G_A = P_6$ b); and through the vertex $\{2\}$ to $G_A = T_4$ c); with the constraint of maximal degree equal to 3.
Table 1.  A sample Matlab code for computing mixed integer semidefinite programming problem (12). The output of the program is the optimal value $\Lambda^{opt}_{HL}(G_A, G_B) = \overline{\Lambda}^{sir}_{HL}(G_A, G_B)$.
mu=sdpvar(1); eta=sdpvar(1); W=intvar(m, m); K=binvar(n, m); ops=sdpsettings('solver', 'bnb', 'bnb.maxiter', bnbmaxiter);
Fconstraints=[...
        [[W, K'];
        [K, eye(n, n)]
        ]>=0, ...
        mu>=0, eta>=0, ...
        [[eye(n, n)-mu*inv(A), K*inv(B)];
        [inv(B)*K', eye(m, m)-mu*inv(B) + inv(B)*W*inv(B)]
        ] >= 0, ...
        [[eye(n, n) + eta*inv(A), K*inv(B)];
               [inv(B)*K', eye(m, m) + eta*inv(B) + inv(B)*W*inv(B)]
        ] >= 0, ...
        sum(K(:, :))==diag(W)', sum(K(:))>=1, ...
        vec(W(:))>=0, 0 < =vec(K(:)) < =1, ...
        sum([[A, K]; [K', B]]) < =maxdegree*ones(1, n+m), ...,
        K*[zeros(kB, m-kB); eye(m-kB, m-kB)] == zeros(n, m-kB), ...
        ];
solvesdp(Fconstraints, -mu-eta, ops)
LambdaSIR = double(mu + eta)
mu=sdpvar(1); eta=sdpvar(1); W=intvar(m, m); K=binvar(n, m); ops=sdpsettings('solver', 'bnb', 'bnb.maxiter', bnbmaxiter);
Fconstraints=[...
        [[W, K'];
        [K, eye(n, n)]
        ]>=0, ...
        mu>=0, eta>=0, ...
        [[eye(n, n)-mu*inv(A), K*inv(B)];
        [inv(B)*K', eye(m, m)-mu*inv(B) + inv(B)*W*inv(B)]
        ] >= 0, ...
        [[eye(n, n) + eta*inv(A), K*inv(B)];
               [inv(B)*K', eye(m, m) + eta*inv(B) + inv(B)*W*inv(B)]
        ] >= 0, ...
        sum(K(:, :))==diag(W)', sum(K(:))>=1, ...
        vec(W(:))>=0, 0 < =vec(K(:)) < =1, ...
        sum([[A, K]; [K', B]]) < =maxdegree*ones(1, n+m), ...,
        K*[zeros(kB, m-kB); eye(m-kB, m-kB)] == zeros(n, m-kB), ...
        ];
solvesdp(Fconstraints, -mu-eta, ops)
LambdaSIR = double(mu + eta)
Table 2.  The computational results and comparison of various semidefinite relaxations. The first two columns describe the graph $G_A$ and $G_B$ with the chosen set of bridging vertices. The optimal value $\Lambda_{HL}^{opt} = \overline{\Lambda}_{HL}^{sir}$ is shown in bold in the middle column. The upper $\overline{\Lambda}_{HL}^{sdp}$ and lower bounds $\underline{\Lambda}_{HL}^{sdp}$, $\underline{\Lambda}_{HL}^{sir}$ are also presented together with computational times in seconds computed on Quad core Intel 1.5GHz CPU with 4 GB of memory.
$G_A$ $G_B$ $\underline{\Lambda}_{HL}^{sdp}$ $\underline{\Lambda}_{HL}^{sir}$ $\Lambda _{HL}^{opt}$=$\bar \Lambda _{HL}^{sir}$ $\overline{\Lambda}_{HL}^{sdp}$ $G_B \mapsto G_A$
$F_0$ $F_0$ $0.233688$ $0.531664$ $\bf 0.74947$ $0.87214$ $1\mapsto 3, 5;\ \ 2\mapsto 6$
$(1, 2)$ $(0.27s)$ $(3.38s)$ $(83s)$ $(2.2s)$
$F_0$ $F_0$ $0.333126$ $0.72678$ $\bf 0.85828$ $0.87214$ $1\mapsto \emptyset;\ \ 4\mapsto 3, 5, 6$
$(1, 4)$ $(0.31s)$ $(4.75s)$ $(36s)$ $(2.2s)$
$F_0$ $F_0$ $0.333126$ $0.719668$ $\bf 0.81389$ $0.87214$ $1\mapsto 4;\ \ 3\mapsto 4$
$(1, 3)$ $(0.31s)$ $(4.27s)$ $(75s)$ $(2.2s)$
$F_1$ $F_0$ $0.163626$ $0.450022$ $\bf 0.56655$ $0.56666$ $1\mapsto \emptyset; \ \ 2\mapsto 9, 11, 12$
$(1, 2)$ $(0.28s)$ $(7.65s)$ $(12470s)$ $(2.2s)$
$P_4$ $P_4$ $0.472136$ $0.86953$ $\bf 1.06418$ $1.23607$ $2\mapsto 2, 4;\ \ 3\mapsto 1, 3$
$(2, 3)$ $(0.27s)$ $(2.18s)$ $(12.6s)$ $(2.2s)$
$P_6$ $P_4$ $0.367365$ $0.811369$ $\bf 0.87366$ $0.89008$ $1\mapsto 4, 6;\ \ 3\mapsto 4, 6$
$(1, 3)$ $(0.26s)$ $(4.6s)$ $(59s)$ $(2.1s)$
$P_6$ $P_4$ $0.367365$ $0.737641$ $\bf 0.87321$ $0.89008$ $2\mapsto 4, 6;\ \ 3\mapsto 1, 3$
$(2, 3)$ $(0.26s)$ $(3.41s)$ $(57s)$ $(2.1s)$
$P_{10}$ $P_4$ $0.252282$ $0.523808$ $\bf 0.56837$ $0.56926$ $2\mapsto 8, 10;\ \ 3\mapsto \emptyset$
$(2, 3)$ $(0.26s)$ $(6.32s)$ $(4109s)$ $(2.6s)$
$T_4$ $P_4$ $0.38832$ $0.73094$ $\bf 0.93258$ $0.95452$ $2\mapsto 3, 8$
$(2)$ $(0.31s)$ $(1.57s)$ $(12s)$ $(2.31s)$
$G_A$ $G_B$ $\underline{\Lambda}_{HL}^{sdp}$ $\underline{\Lambda}_{HL}^{sir}$ $\Lambda _{HL}^{opt}$=$\bar \Lambda _{HL}^{sir}$ $\overline{\Lambda}_{HL}^{sdp}$ $G_B \mapsto G_A$
$F_0$ $F_0$ $0.233688$ $0.531664$ $\bf 0.74947$ $0.87214$ $1\mapsto 3, 5;\ \ 2\mapsto 6$
$(1, 2)$ $(0.27s)$ $(3.38s)$ $(83s)$ $(2.2s)$
$F_0$ $F_0$ $0.333126$ $0.72678$ $\bf 0.85828$ $0.87214$ $1\mapsto \emptyset;\ \ 4\mapsto 3, 5, 6$
$(1, 4)$ $(0.31s)$ $(4.75s)$ $(36s)$ $(2.2s)$
$F_0$ $F_0$ $0.333126$ $0.719668$ $\bf 0.81389$ $0.87214$ $1\mapsto 4;\ \ 3\mapsto 4$
$(1, 3)$ $(0.31s)$ $(4.27s)$ $(75s)$ $(2.2s)$
$F_1$ $F_0$ $0.163626$ $0.450022$ $\bf 0.56655$ $0.56666$ $1\mapsto \emptyset; \ \ 2\mapsto 9, 11, 12$
$(1, 2)$ $(0.28s)$ $(7.65s)$ $(12470s)$ $(2.2s)$
$P_4$ $P_4$ $0.472136$ $0.86953$ $\bf 1.06418$ $1.23607$ $2\mapsto 2, 4;\ \ 3\mapsto 1, 3$
$(2, 3)$ $(0.27s)$ $(2.18s)$ $(12.6s)$ $(2.2s)$
$P_6$ $P_4$ $0.367365$ $0.811369$ $\bf 0.87366$ $0.89008$ $1\mapsto 4, 6;\ \ 3\mapsto 4, 6$
$(1, 3)$ $(0.26s)$ $(4.6s)$ $(59s)$ $(2.1s)$
$P_6$ $P_4$ $0.367365$ $0.737641$ $\bf 0.87321$ $0.89008$ $2\mapsto 4, 6;\ \ 3\mapsto 1, 3$
$(2, 3)$ $(0.26s)$ $(3.41s)$ $(57s)$ $(2.1s)$
$P_{10}$ $P_4$ $0.252282$ $0.523808$ $\bf 0.56837$ $0.56926$ $2\mapsto 8, 10;\ \ 3\mapsto \emptyset$
$(2, 3)$ $(0.26s)$ $(6.32s)$ $(4109s)$ $(2.6s)$
$T_4$ $P_4$ $0.38832$ $0.73094$ $\bf 0.93258$ $0.95452$ $2\mapsto 3, 8$
$(2)$ $(0.31s)$ $(1.57s)$ $(12s)$ $(2.31s)$
Table 3.  The computational results and comparison of various relaxations. The chosen graphs and description of columns is the same as in Table 2. In this table we present results of optimization when additional constraint of the maximal degree 3 has been imposed.
$G_A$ $G_B$ $\underline{\Lambda}_{HL}^{sdp}$ $\underline{\Lambda}_{HL}^{sir}$ $\Lambda_{HL}^{opt}=\overline{\Lambda}_{HL}^{sir}$ $\overline{\Lambda}_{HL}^{sdp}$ $G_B \mapsto G_A$
$F_0$ $F_0$ $0.233688$ $0.507678$ $\bf 0.720830$ $0.87214$ $1\mapsto \emptyset;\ 2\mapsto 6$
$(1, 2)$ $(0.31s)$ $(2.73s)$ $(7.1s)$ $(2.9s)$
$F_0$ $F_0$ $0.233688$ $0.468053$ $\bf0.720830$ $0.87214$ $1\mapsto6;4\mapsto\emptyset$
$(1, 4)$ $(0.31s)$ $(1.1s)$ $(2.33s)$ $(2.85s)$
$F_0$ $F_0$ $0.333126$ $0.706635$ $\bf0.776875$ $0.87214$ $1\mapsto6;3\mapsto6$
$(1, 3)$ $(0.35s)$ $(2.45s)$ $(8.4s)$ $(2.82s)$
$F_1$ $F_0$ $0.163626$ $0.389941$ $\bf0.493727$ $0.566658$ $1\mapsto6;2\mapsto\emptyset$
$(1, 2)$ $(0.38s)$ $(3.67s)$ $(13.4s)$ $(2.83s)$
$P_4$ $P_4$ $0.472136$ $0.869530$ $\bf0.954520$ $1.23607$ $3\mapsto\emptyset;2\mapsto2$
$(2, 3)$ $(0.31s)$ $(1.86s)$ $(7.8s)$ $(2.86s)$
$P_6$ $P_4$ $0.367365$ $0.811369$ $\bf0.828427$ $0.89008$ $1\mapsto4, 6;3\mapsto2$
$(1, 3)$ $(0.36s)$ $(3.35s)$ $(22.9s)$ $(2.83s)$
$P_6$ $P_4$ $0.367365$ $0.737641$ $\bf0.820751$ $0.89008$ $2\mapsto5;3\mapsto2$
$(2, 3)$ $(0.33)$ $(2.73s)$ $(9.21s)$ $(2.87s)$
$P_{10}$ $P_4$ $0.252282$ $0.523808$ $\bf0.559046$ $0.56926$ $2\mapsto\emptyset;3\mapsto11$
$(2, 3)$ $(0.33s)$ $(4.78s)$ $(13.87s)$ $(2.86s)$
$T_4$ $P_4$ $0.38832$ $0.692266$ $\bf0.890084$ $0.95452$ $2\mapsto4$
$(2)$ $(0.31s)$ $(0.88s)$ $(1.5s)$ $(2.11s)$
$G_A$ $G_B$ $\underline{\Lambda}_{HL}^{sdp}$ $\underline{\Lambda}_{HL}^{sir}$ $\Lambda_{HL}^{opt}=\overline{\Lambda}_{HL}^{sir}$ $\overline{\Lambda}_{HL}^{sdp}$ $G_B \mapsto G_A$
$F_0$ $F_0$ $0.233688$ $0.507678$ $\bf 0.720830$ $0.87214$ $1\mapsto \emptyset;\ 2\mapsto 6$
$(1, 2)$ $(0.31s)$ $(2.73s)$ $(7.1s)$ $(2.9s)$
$F_0$ $F_0$ $0.233688$ $0.468053$ $\bf0.720830$ $0.87214$ $1\mapsto6;4\mapsto\emptyset$
$(1, 4)$ $(0.31s)$ $(1.1s)$ $(2.33s)$ $(2.85s)$
$F_0$ $F_0$ $0.333126$ $0.706635$ $\bf0.776875$ $0.87214$ $1\mapsto6;3\mapsto6$
$(1, 3)$ $(0.35s)$ $(2.45s)$ $(8.4s)$ $(2.82s)$
$F_1$ $F_0$ $0.163626$ $0.389941$ $\bf0.493727$ $0.566658$ $1\mapsto6;2\mapsto\emptyset$
$(1, 2)$ $(0.38s)$ $(3.67s)$ $(13.4s)$ $(2.83s)$
$P_4$ $P_4$ $0.472136$ $0.869530$ $\bf0.954520$ $1.23607$ $3\mapsto\emptyset;2\mapsto2$
$(2, 3)$ $(0.31s)$ $(1.86s)$ $(7.8s)$ $(2.86s)$
$P_6$ $P_4$ $0.367365$ $0.811369$ $\bf0.828427$ $0.89008$ $1\mapsto4, 6;3\mapsto2$
$(1, 3)$ $(0.36s)$ $(3.35s)$ $(22.9s)$ $(2.83s)$
$P_6$ $P_4$ $0.367365$ $0.737641$ $\bf0.820751$ $0.89008$ $2\mapsto5;3\mapsto2$
$(2, 3)$ $(0.33)$ $(2.73s)$ $(9.21s)$ $(2.87s)$
$P_{10}$ $P_4$ $0.252282$ $0.523808$ $\bf0.559046$ $0.56926$ $2\mapsto\emptyset;3\mapsto11$
$(2, 3)$ $(0.33s)$ $(4.78s)$ $(13.87s)$ $(2.86s)$
$T_4$ $P_4$ $0.38832$ $0.692266$ $\bf0.890084$ $0.95452$ $2\mapsto4$
$(2)$ $(0.31s)$ $(0.88s)$ $(1.5s)$ $(2.11s)$
[1]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[2]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[3]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[4]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[5]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[6]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[7]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[8]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[9]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

 Impact Factor: 

Metrics

  • PDF downloads (65)
  • HTML views (432)
  • Cited by (0)

Other articles
by authors

[Back to Top]