关于图的能量及扩展能量
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摘要
图G的能量是指图G的邻接矩阵特征值的绝对值之和﹒简要介绍近几年来国内外学者对能量以及扩展能量的研究情况和他们所取得的成果;重点介绍了几类扩展能量,譬如预解能量、塞德尔能量、埃尔米特能量以及斜能量的研究成果;同时提出了在能量研究中存在的某些问题以及今后需要努力的一些方向﹒
The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. This paper introduces the research situation and achievements on energy and extended erengy of graphs at home and abroad in the past few years, it focuses on a few class extended energy, such as resolvent energy, Seidel energy, Hermitian energy and skew energy. At the same time the author also puts forward some existing problems, as well as pointing out some direction in the future.
The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. This paper introduces the research situation and achievements on energy and extended erengy of graphs at home and abroad in the past few years, it focuses on a few class extended energy, such as resolvent energy, Seidel energy, Hermitian energy and skew energy. At the same time the author also puts forward some existing problems, as well as pointing out some direction in the future.
引文
[1]GUTMAN I.The energy of a graph[J].Ber Math-Statist Sekt Forsch Graz,1978,103:2177-2187.
[2]GUTMAN I,RADENKOVI?S,DORDEVI?S,et al.Totalπ-electron and HOMO energy[J].Chemical Physics Letters,2016,649:148-150.
[3]GUTMAN I.Totalπ-electron energy of conjugated molecules with non-bonding molecular orbitals[J].Zeitschrift für Naturforschung A,2016,71(2):161-164.
[4]GUTMAN I,RADENKOVI?S,DORDEVI?S,et al.Extending the Mc Clelland formula for totalπ-electron energy[J].Journal of Mathematical Chemistry,2017,55(10):1934-1940.
[5]MILOVANOVI?I,MILOVANOVI?E,GUTMAN I.Upper bounds for some graph energies[J].Applied Mathematics and Computation,2016,289:435-443.
[6]GUTMAN I,FURTULA B.Survey of graph energies[J].Mathematics Interdisciplinary Research,2017,2:85-129.
[7]LI X L,SHI Y T,GUTMAN I.Graph energy[M].New York,Springer,2012.
[8]GUTMAN I,ZHOU B.Laplacian energy of a graph[J].Linear Algebra and its Applications,2006,414(1):29-37.
[9]ABREU N,CARDOSO D M,GUTMAN I,et al.Bounds for the signless Laplacian energy[J].Linear Algebra and its Applications,2011,435(10):2365-2374.
[10]INDULAL G,GUTMAN I,VIJAYAKUMAR A.On distance energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2008,60(2):461-472.
[11]RAMANE H S,REVANKAR D S,GUTMAN I,et al.Bounds for the distance energy of a graph[J].Kragujevac Journal of Mathematics,2008,31:59-68.
[12]ILI?A.Distance spectra and distance energy of integral circulant graphs[J].Linear Algebra and its Applications,2010,433(5):1005-1014.
[13]GüNG?R A D,BOZKURT?B.On the distance spectral radius and the distance energy of graphs[J].Linear and Multilinear Algebra,2011,59(4):365-370.
[14]LU J P,LIU B L.A Laplacian-energy-like invariant of a graph[J].MATCH Communications in Mathematical and in Computer Chemistry,2008,59(2):355-372.
[15]ZHU B X.The Laplacian-energy like of graphs[J].Applied Mathematics Letters,2011,24(9):1604-1607.
[16]WANG W Z,LUO Y F.On Laplacian-energy-like invariant of a graph[J].Linear Algebra and its Applications,2012,437(2):713-721.
[17]XU K X,DAS K C.Extremal Laplacian-energy-like invariant of graphs with given matching number[J].The Electronic Journal of Linear Algebra,2013,26:131-140.
[18]PIRZADA S,GANIE H A,GUTMAN I.On Laplacian-energylike invariant and Kirchhoff index[J].MATCH Communications in Mathematical and in Computer Chemistry,2015,73:41-59.
[19]DE FREITAS M A A,GUTMAN I,ROBBIANO M.Graphs with maximum Laplacian-energy-like invariant and incidence energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:331-342.
[20]BOZKURT?B,GüNG?R A D,GUTMAN I,et al.Randi?matrix and Randi?energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2010,64(1):239-250.
[21]BOZKURT?B,GüNG?R A D,GUTMAN I.Randi?spectral radius and Randi?energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2010,64(2):321-334.
[22]GUTMAN I,MARTINS E A,ROBBIANO M,et al.Ky Fan theorem applied to Randi?energy[J].Linear Algebra and its Applications,2014,459:23-42.
[23]GUTMAN I,ROBBIANO M,MARTIN B S.Upper bound on Randi?energy of some graphs[J].Linear Algebra and its Applications,2015,478:241-255.
[24]DAS K C,SUN S W,GUTMAN I.Normalized Laplacian eigenvalues and Randi?energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2017,77:45-59.
[25]HAEMERS W H.Seidel switching and graph energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:653-659.
[26]RAMANE H S,GUTMAN I,GUNDLOOR M M.Seidel energy of iterated line graphs of regular graphs[J].Kragujevac Journal of Mathematics,2015,39(1):7-12.
[27]OBOUDI M R.Energy and Seidel energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:291-303.
[28]GUTMAN I,FURTULA B,ZOGI?E,et al.Resolvent energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:279-290.
[29]ALLEM L E,CAPAVERDE J,TREVISAN V,et al.Resolvent energy of unicyclic,bicyclic and tricyclic graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2017,77:95-104.
[30]ZHU Z X.Some extremal properties of the resolvent energy,Estrada and resolvent Estrada indices of graphs[J].Journal of Mathematical Analysis and Applications,2017,447(2):957-970.
[31]GUTMAN I,KIANI D,MIRZAKHAH M,et al.On incidence energy of a graph[J].Linear Algebra and its Applications,2009,431(8):1223-1233.
[32]JOOYANDEH M,KIANI D,MIRZAKHAH M.Incidence energy of a graph[J].MATCH Communications in Mathematical and in Computer Chemistry,2009,62(3):561-572.
[33]DAS K C,GUTMAN I.On incidence energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2014,446:329-344.
[34]ZHANG J B,LI J P.New results on the incidence energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:777-803.
[35]TANG Z K,HOU Y P.On incidence energy of trees[J].MATCH Communications in Mathematical and in Computer Chemistry,2011,66:977-984.
[36]ROJO O,LENES E.A sharp upper bound on the incidence energy of graphs in terms of connectivity[J].Linear Algebra and its Applications,2013,438(3):1485-1493.
[37]ZHANG J B,KAN H B,LIU X D.Graphs with extremal incidence energy[J].Filomat,2015,29(6):1251-1258.
[38]LIU J X,LI X L.Hermitian-adjacency matrices and Hermitian energies of mixed graphs[J].Linear Algebra and its Applications,2015,466:182-207.
[39]CHEN X L,LI X L,ZHANG Y Y.3-Regular mixed graphs with optimum Hermitian energy[J].Linear Algebra and its Applications,2016,496:475-486.
[40]ADIGA C,BALAKRISHNAN R,WASIN SO.The skew energy of a digraph[J].Linear Algebra and its Applications,2010,432(7):1825-1835.
[41]HOU Y P,SHEN X L,ZHANG C Y.Oriented unicyclic graphs with extremal skew energy[J/OL].https://arxiv.org/pdf/1108.6229v1.pdf
[42]ZHU J M.Oriented unicyclic graphs with the first?(n-9)2???largest skew energies[J].Linear Algebra and its Applications,2012,437(10):2630-2649.
[43]SHEN X L,HOU Y P,ZHANG C Y.Bicyclic digraphs with extremal skew energy[J].Electronic Journal of Linear Algebra,2012,23:340-355.
[44]GONG S C,XU G H.3-Regular digraphs with optimum skew energy[J].Linear Algebra and its Applications,2012,436(3):465-471.
[45]CHEN X L,LI X L,LIAN H S.4-Regular oriented graphs with optimum skew energy[J].Linear Algebra and its Applications,2013,439(10):2948-2960.
[46]GUTMAN I,VIDOVI?D,CMILJANOVI?N,et al.Graph energy—A useful molecular structure-descriptor[J].Indian Journal of Chemistry A,2003,42:1309-1311.
[47]DEHMER M,LI X L,SHI Y T.Connections between generalized graph entropies and graph energy[J].Complexity,2015,21(1):35-41.
[48]DAIANU M,MEZHER A,JAHANSHAD N,et al.Spectral graph theory and graph energy metrics show evidence for the Alzheimer’s disease disconnection syndrome in APOE-4 risk gene carriers[C].2015 IEEE 12th International Symposium on Biomedical Imaging(ISBI),New York,2015:458-461.
[49]VAN MIEGHEM P,VAN DE BOVENKAMP R.Accuracy criterion for the mean-field approximation in susceptibleinfected-susceptible epidemics on networks[J].Physical review E,Statistical,Nonlinear,and Soft Matter Physics,2015,91(3):(032812)1-11.
[2]GUTMAN I,RADENKOVI?S,DORDEVI?S,et al.Totalπ-electron and HOMO energy[J].Chemical Physics Letters,2016,649:148-150.
[3]GUTMAN I.Totalπ-electron energy of conjugated molecules with non-bonding molecular orbitals[J].Zeitschrift für Naturforschung A,2016,71(2):161-164.
[4]GUTMAN I,RADENKOVI?S,DORDEVI?S,et al.Extending the Mc Clelland formula for totalπ-electron energy[J].Journal of Mathematical Chemistry,2017,55(10):1934-1940.
[5]MILOVANOVI?I,MILOVANOVI?E,GUTMAN I.Upper bounds for some graph energies[J].Applied Mathematics and Computation,2016,289:435-443.
[6]GUTMAN I,FURTULA B.Survey of graph energies[J].Mathematics Interdisciplinary Research,2017,2:85-129.
[7]LI X L,SHI Y T,GUTMAN I.Graph energy[M].New York,Springer,2012.
[8]GUTMAN I,ZHOU B.Laplacian energy of a graph[J].Linear Algebra and its Applications,2006,414(1):29-37.
[9]ABREU N,CARDOSO D M,GUTMAN I,et al.Bounds for the signless Laplacian energy[J].Linear Algebra and its Applications,2011,435(10):2365-2374.
[10]INDULAL G,GUTMAN I,VIJAYAKUMAR A.On distance energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2008,60(2):461-472.
[11]RAMANE H S,REVANKAR D S,GUTMAN I,et al.Bounds for the distance energy of a graph[J].Kragujevac Journal of Mathematics,2008,31:59-68.
[12]ILI?A.Distance spectra and distance energy of integral circulant graphs[J].Linear Algebra and its Applications,2010,433(5):1005-1014.
[13]GüNG?R A D,BOZKURT?B.On the distance spectral radius and the distance energy of graphs[J].Linear and Multilinear Algebra,2011,59(4):365-370.
[14]LU J P,LIU B L.A Laplacian-energy-like invariant of a graph[J].MATCH Communications in Mathematical and in Computer Chemistry,2008,59(2):355-372.
[15]ZHU B X.The Laplacian-energy like of graphs[J].Applied Mathematics Letters,2011,24(9):1604-1607.
[16]WANG W Z,LUO Y F.On Laplacian-energy-like invariant of a graph[J].Linear Algebra and its Applications,2012,437(2):713-721.
[17]XU K X,DAS K C.Extremal Laplacian-energy-like invariant of graphs with given matching number[J].The Electronic Journal of Linear Algebra,2013,26:131-140.
[18]PIRZADA S,GANIE H A,GUTMAN I.On Laplacian-energylike invariant and Kirchhoff index[J].MATCH Communications in Mathematical and in Computer Chemistry,2015,73:41-59.
[19]DE FREITAS M A A,GUTMAN I,ROBBIANO M.Graphs with maximum Laplacian-energy-like invariant and incidence energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:331-342.
[20]BOZKURT?B,GüNG?R A D,GUTMAN I,et al.Randi?matrix and Randi?energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2010,64(1):239-250.
[21]BOZKURT?B,GüNG?R A D,GUTMAN I.Randi?spectral radius and Randi?energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2010,64(2):321-334.
[22]GUTMAN I,MARTINS E A,ROBBIANO M,et al.Ky Fan theorem applied to Randi?energy[J].Linear Algebra and its Applications,2014,459:23-42.
[23]GUTMAN I,ROBBIANO M,MARTIN B S.Upper bound on Randi?energy of some graphs[J].Linear Algebra and its Applications,2015,478:241-255.
[24]DAS K C,SUN S W,GUTMAN I.Normalized Laplacian eigenvalues and Randi?energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2017,77:45-59.
[25]HAEMERS W H.Seidel switching and graph energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:653-659.
[26]RAMANE H S,GUTMAN I,GUNDLOOR M M.Seidel energy of iterated line graphs of regular graphs[J].Kragujevac Journal of Mathematics,2015,39(1):7-12.
[27]OBOUDI M R.Energy and Seidel energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:291-303.
[28]GUTMAN I,FURTULA B,ZOGI?E,et al.Resolvent energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:279-290.
[29]ALLEM L E,CAPAVERDE J,TREVISAN V,et al.Resolvent energy of unicyclic,bicyclic and tricyclic graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2017,77:95-104.
[30]ZHU Z X.Some extremal properties of the resolvent energy,Estrada and resolvent Estrada indices of graphs[J].Journal of Mathematical Analysis and Applications,2017,447(2):957-970.
[31]GUTMAN I,KIANI D,MIRZAKHAH M,et al.On incidence energy of a graph[J].Linear Algebra and its Applications,2009,431(8):1223-1233.
[32]JOOYANDEH M,KIANI D,MIRZAKHAH M.Incidence energy of a graph[J].MATCH Communications in Mathematical and in Computer Chemistry,2009,62(3):561-572.
[33]DAS K C,GUTMAN I.On incidence energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2014,446:329-344.
[34]ZHANG J B,LI J P.New results on the incidence energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:777-803.
[35]TANG Z K,HOU Y P.On incidence energy of trees[J].MATCH Communications in Mathematical and in Computer Chemistry,2011,66:977-984.
[36]ROJO O,LENES E.A sharp upper bound on the incidence energy of graphs in terms of connectivity[J].Linear Algebra and its Applications,2013,438(3):1485-1493.
[37]ZHANG J B,KAN H B,LIU X D.Graphs with extremal incidence energy[J].Filomat,2015,29(6):1251-1258.
[38]LIU J X,LI X L.Hermitian-adjacency matrices and Hermitian energies of mixed graphs[J].Linear Algebra and its Applications,2015,466:182-207.
[39]CHEN X L,LI X L,ZHANG Y Y.3-Regular mixed graphs with optimum Hermitian energy[J].Linear Algebra and its Applications,2016,496:475-486.
[40]ADIGA C,BALAKRISHNAN R,WASIN SO.The skew energy of a digraph[J].Linear Algebra and its Applications,2010,432(7):1825-1835.
[41]HOU Y P,SHEN X L,ZHANG C Y.Oriented unicyclic graphs with extremal skew energy[J/OL].https://arxiv.org/pdf/1108.6229v1.pdf
[42]ZHU J M.Oriented unicyclic graphs with the first?(n-9)2???largest skew energies[J].Linear Algebra and its Applications,2012,437(10):2630-2649.
[43]SHEN X L,HOU Y P,ZHANG C Y.Bicyclic digraphs with extremal skew energy[J].Electronic Journal of Linear Algebra,2012,23:340-355.
[44]GONG S C,XU G H.3-Regular digraphs with optimum skew energy[J].Linear Algebra and its Applications,2012,436(3):465-471.
[45]CHEN X L,LI X L,LIAN H S.4-Regular oriented graphs with optimum skew energy[J].Linear Algebra and its Applications,2013,439(10):2948-2960.
[46]GUTMAN I,VIDOVI?D,CMILJANOVI?N,et al.Graph energy—A useful molecular structure-descriptor[J].Indian Journal of Chemistry A,2003,42:1309-1311.
[47]DEHMER M,LI X L,SHI Y T.Connections between generalized graph entropies and graph energy[J].Complexity,2015,21(1):35-41.
[48]DAIANU M,MEZHER A,JAHANSHAD N,et al.Spectral graph theory and graph energy metrics show evidence for the Alzheimer’s disease disconnection syndrome in APOE-4 risk gene carriers[C].2015 IEEE 12th International Symposium on Biomedical Imaging(ISBI),New York,2015:458-461.
[49]VAN MIEGHEM P,VAN DE BOVENKAMP R.Accuracy criterion for the mean-field approximation in susceptibleinfected-susceptible epidemics on networks[J].Physical review E,Statistical,Nonlinear,and Soft Matter Physics,2015,91(3):(032812)1-11.