几类特殊图的广义邻接距离谱.pdf

Advances in Applied Mathematics A?,2024,13(4),1542-1557Published Online April 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.134145AaA?2?l“?,7uvF2024c3?23FF2024c4?21FuF2024c4?28F nCc5?22?CPastWnRojo?#?2?l?5|2?l?d?4Kn?-K?Aa?4a5X?oaA?2?l=?l?1?2?l?1?2c?y,?,2?l?Generalized Adjacency DistanceSpectrum of Several Classes ofSpecial GraphsHuilu SunSchool of Mathematical Sciences,Zhejiang Normal University,Jinhua ZhejiangReceived:Mar.23rd,2024;accepted:Apr.21st,2024;published:Apr.28th,2024:.AaA?2?lJ.A?,2024,13(4):1542-1557.DOI:10.12677/aam.2024.134145AbstractGraph theory has been widely studied by many scholars in recent years,among whichthe study on generalized matrix has attracted many scholars.Recently,Pasten andRojo introduced a new generalized matrix,which is a convex linear combination of theadjacency matrix and the distance matrix,called the generalized adjacency distancematrix.In addition,extremal spectrum is also an important subject in spectrumtheory.Some special graphs are often studied as extremal graphs,such as blockindifference graphs,pineapple graphs,etc.In this paper,we study the generalizedadjacency distance spectrum of four kinds of special graphs.We generalize not onlythe adjacency spectrum and distance spectrum,but also their adjacency distancespectrum.KeywordsEqual Division,Spectrum of a Graph,Generalized Adjacency Distance MatrixCopyright c?2024 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.nCc5?2,2?.32017c,Nikiforov 1?G?2?A(G)?:A(G)=De(G)+(1 )A(G),0 1,G?A?.w,2?A(G)?A(G)?De(G)?5|.32022 c,Brondani?3z 2?oaA:Kqp,?,KKlnHln?AA?,?A(G)?Iv?.C,PastWn Rojo 3z?#?2?,?A(G)l?D(G)?5|,2?l?,S(G)L,S(G)=D(G)+(1 )A(G).?=12,S12(G)=12S(G),S(G)=A(G)+D(G)G?l?.u?l?!z 3.DOI:10.12677/aam.2024.1341451543A?d?,4Kn?-K.?Aa?4a5,X:?,Hln,KqpKKln(z 47).d,?AaA?2?l?1?x.dd=2?l?(J,2?l?(J.2.?!0?I?Vg,B?Y.?.G,?n,-A(G)De(G)O“LG?.G:v w?l?v w:?,dvw5L.2.1.8-A n?,1?I8V=1,2,.,n L.XJV1,V2,.,Vk 8V?y|Vi|=ni,i=1,2,.,k,oA?Xe?:A=A1,1A1,2A1,kA2,1A2,2A2,k.Ak,1Ak,2Ak,k,Ai,j“LA?ni nj?f?,1 i,j k.-bi,j“Lf?Ai,j?1,o?B=(bi,j)?A?.d?,XJzf?Ai,j?1?,oy?y,A?B?A?.2.2.9-G0(n1,n2,.,nk)k?,k?O:n1,n2,.,nk,ni 2,1 i k.d?,G0(n1,n2,.,nk)k?Xe:(1)G0(n1)=Kn1;(2)G0(n1,n2)dKn1Kn2:?;(3)G0(n1,n2,.,nt)dG0(n1,n2,.,nt1)?Knt1Knt:?,t?,X2 H36.2.4.2-q 1,p 3,oKqpL3?Kp?:!q?,X3(a)K35.DOI:10.12677/aam.2024.1341451544A?Figure 1.R(G)=3,4,5 1.R(G)=3,4,5Figure 2.H36 2.H362.5.2-n 3,1 l n,oKKlnL3?Kn?:,?Kn?l:mOl?,X3(b)KK35.Figure 3.K35and KK35 3.K35KK35n2.6.8 XJ?B?A?y?,o?B?kA?A?A?.n2.7.10 XJ?P=C1C2C2C1#?,o?P?A?=?C1+C2?C1 C2?A?.3.AaA?2?l?!?,Hln,KqpKKln?2?l./?y,?9?AaA?2?l,=?l?1?2,DOI:10.12677/aam.2024.1341451545A?l?1?2.d?,?Y?l4KC?:.3.1.?2?l?!|?y?x?2?l.T(J2?z 11 ff?(=?)?(J,?2?l?l?(J.n3.1.-G=G0(n1,n2)?,0,1),n1 n2,o?G?SA?1 9fs1(x)?n,A?1?-n1+n2 4,fs1(x)=x3+(n1+n24)x2+42(n11)(n21)+3n1+3n2n1n26x+4(n11)(n21)+3n1+3n22n1n24.y.LG?:?I,?2?l?S(G)Xe:Jn11 In11J(n11)12J(n11)(n21)0.3cmJ1(n11)0J1(n21)2J(n21)(n11)J(n21)1Jn21 In21.S(G)+In1+n21?cn11 1,?n21 1,0 S(G)+In1+n21?A?-?n1+n2 4.X1 S(G)?A?-?n1+n2 4.uS(G)?A?,d?Bs1?.y3,?Bs1=n1 212(n2 1)n1 10n2 12(n1 1)1n2 2.dn2.6?,Bs1?A?S(G)?A?.LO,Bs1?A?fs1(x).?n?y.3.2.?,kXe(J:(1)?=0,S(G)=A(G).d,?2?lA?=?A?,z 11?n1(J.(2)?=12,S(G)=S(G).d,?2?lA?=?lA?.(3)?=1,S(G)=D(G).d,?2?lA?=lA?.du2?l?l?5|,LOG?l5?yG?2?l?(5.3.3.?R(G)=3,4,X4.DOI:10.12677/aam.2024.1341451546A?Figure 4.R(G)=3,4 4.R(G)=3,4K?l?O:A(R(G)=011000101000110111001011001101001110,D(R(G)=011222101222110111221011221101221110,KR(G)?l?:S(R(G)=022222202222220222222022222202222220.LO,?A(R(G)?A?:fsA(x)=(x+1)3(x3 3x2 3x+7),DOI:10.12677/aam.2024.1341451547A?l?D(R(G)?A?:fsD(x)=(x+1)3(x3 3x2 27x 17).q?=0,n1=3,n2=4,fs1(x)=x3+3x2+3x 7.d,dn3.1,?R(G)=3,4?2?lA?1 9fs1(x)=x3+3x2+3x7?n,A?1?-3.?/,?=1,n1=3,n2=4,fs1(x)=x3+3x2+27x+17.d,dn3.1,?R(G)=3,4?2?lA?1 9fs1(x)=x3+3x2+27x+17?n,A?1?-3.L*?,12S(R(G)?:5(1),1(5).q?=12,n1=3,n2=4,fs1(x)=x3+3x2+9x+5=(x+1)2(5x).d,dn3.1,?R(G)=3,4?2?l:5(1),1(5).n,?=0,n1=3,n2=4,?R(G)=3,4?2?l?;?=1,n1=3,n2=4,?R(G)=3,4?2?ll?,?=12,?R(G)=3,4?2?l?l?12?.3.2.Hln?2?ln3.4.-0,1)G Hln.oG?SA?2,2 2,1 9s1,s2,s1,s2,-Ol1,l1,2(nl1),1,1,1,1.s1s2ms1(x)?,s1s2ms2(x)?,ms1(x)=x2+1n+(l3n+2)x+2(6l2l26n+2ln)+(2l22n+ln)l,ms2(x)=x2+3n+(3nl2)x+2(6l2l26n+2ln)+(26l+l2+8nln)+l+22n.y.-G Hln.LG?:?I,?G?2?l?S(G)=EsFsFsEs#,Es=Jl IlJl(nl)J(nl)lJnl Inl,Fs=2Jl+(1 2)Il2Jl(nl)2J(nl)l3Jnl.dn2.7?,S(G)?A?Es+FsEs FsA?.?Es+Fs=(2+1)Jl 2Il(2+1)Jl(nl)(2+1)J(nl)l(3+1)Jnl Inl,Es Fs=(1 2)Jl+(2 2)Il(1 2)Jl(nl)(1 2)J(nl)l(1 3)Jnl Inl.DOI:10.12677/aam.2024.1341451548A?aqn3.1?y,2 1 Es+Fs?A?-?l 1 n l 1.uEs+Fs?A?d?Ms1?.N?Ms1=(2+1)(l 1)+1(2+1)(n l)(2+1)l(3+1)(n l)1.dn2.6,Ms1?A?Es+Fs?A?.LO,?Ms1?A?ms1(x)=x2+1 n+(l 3n+2)x+2(6l 2l2 6n+2ln)+(2 l2 2n+ln)l.-s1s2ms1(x)?,KEs+Fs?2(l1),1(nl1),s1,s2.?/,2 2 1 Es Fs?A?-?l 1 n l 1.uEs FsA?d?Ms2?.N?Ms2=(1 2)(l 1)1(1 2)(n l)(1 2)l(1 3)(n l)1,Kdn2.6,Ms2?A?Es Fs?A?.LO,?Ms2?A?ms2(x)=x2+3n+(3nl2)x+2(6l2l26n+2ln)+(2 6l+l2+8n ln)+l+2 2n.-s1s2ms2(x)?,oEs Fs?(2 2)(l1),1(nl1),s1,s2.?,dn2.7,?n?(J.3.5.?,kXe(J:(1)?=0,S(G)=A(G).d,Hln?2?lA?=?A?.(2)?=12,S(G)=S(G).d,Hln?2?lA?=?lA?.(3)?=1,S(G)=D(G).d,Hln?2?lA?=lA?.3.6.H35,X5.Figure 5.H35 5.H35DOI:10.12677/aam.2024.1341451549A?K?l?O:A(H35)=0111110000101110100011011001001110100000111100000010000011110100010111001001101100000111010000011110,D(H35)=0111112222101112122211011221221110122233111102223312222011112122210111221221101122233111012223311110,oH35?l?S(H35)=0222222222202222222222022222222220222233222202223322222022222222220222222222202222233222022223322220.LO,?A(H35)?A?:fsA(x)=x2(x+1)2(x+2)2(x24x3)(x2DOI:10.12677/aam.2024.1341451550A?2x5),l?D(H35)?A?:fsD(x)=x2(x+1)2(x+2)2(x214x5)(x2+8x+9),?l?S(H35)?A?fsAD(x)=(x+2)7(x+4)(x2 18x 16),?12S(H35)?:(1)(7),2,9+972,9972.q?=0,n=5,l=3,ms1(x)=x2 4x 3,ms2(x)=x2 2x 5.d,dn3.4,H35?2?lA?0,2,1 9ms1(x)=x2 4x 3?ms2(x)=x2 2x 5?,A?0,2,1?-2.?/,?=1,n=5,l=3,ms1(x)=x2 14x 5,ms2(x)=x2+8x+9.d,dn3.4,H35?2?lA?2,0,1 9ms1(x)=x2 14x 5?ms2(x)=x2+8x+9?,A?0,2,1?-2.?=12,n=5,l=3,ms1(x)=x29x4,ms2(x)=x2+3x+2.d,dn3.4,H35?2?lA?1,2 99+972,9972,-O:7,1,1,1.n,?=0,n=5,l=3,H35?2?lH35?;?=1,n=5,l=3,H35?2?lH35?l?,?=12,n=5,l=3,H35?2?lH35?l?12?.3.3.Kqp?2?l?!|?y?x?Kqp?2?l.T(J2?z 12 Kqp?z 13 Kqpl?(J,?2?Kqp?l?(J.n3.7.-0,1),G Kqpn=p+q,KG?SA?1,2 9fs2(x)?n.A?1,2?-Op 2,q 1,fs2(x)=x3+p 2+2(q 1)x2+p 1 2(p 2)(q 1)+q+42(p 1)qx+2(p 1)(q+1)(p 2)q.y.LG?:?I,G?2?l?S(G)LXe:0J1(p1)J1qJ(p1)1Jp1 Ip12J(p1)qJq12Jq(p1)2Jq 2Iq.S(G)+Inkq+2?1,0 S(G)+In?A?-?p 2.X1 S(G)?A?-?p 2.aq/,S(G)+2Inkp+1?1,X2 S(G)?A?-?q 1.uS(G)?A?,d?Bs2?.?,Bs2=0p 1q1p 22q12(p 1)2(q 1).DOI:10.12677/aam.2024.1341451551A?dn2.6,Bs2?A?S(G)?A?.LO,Bs2?A?fs2(x).?n?y.3.8.?,kXe(J:(1)?=0,S(G)=A(G).d,Kqp?2?lA?=?A?,z 12?K1.1(J.(2)?=12,S(G)=S(G).d,Kqp?2?lA?=?lA?.(3)?=1,S(G)=D(G).d,Kqp?2?lA?=lA?,z 13?n2.3(J.3.9.K46,X6.Figure 6.K46 6.K46K?l?O:A(K46)=0111111111101111000011011100001110110000111101000011111000001000000000100000000010000000001000000000,DOI:10.12677/aam.2024.1341451552A?D(K46)=0111111111101111222211011122221110112222111101222211111022221222220222122222202212222222021222222220,oK46?l?:S(K46)=0222222222202222222222022222222220222222222202222222222022222222220222222222202222222222022222222220.LO,?A(K46)?A?:fsA(x)=x3(x+1)4(x3 4x2 9x+16),l?D(K46)?A?:fsD(x)=(x+1)4(x+2)3(x3 10 x2 65x 34).q?=0,p=6,q=4,fs2(x)=x3+4x2+9x 16.d,dn3.7,K46?2?lA?1,0 9fs2(x)=x3+4x2+9x 16?n,A?10?-O4 3.?/,?=1,p=6,q=4,fs2(x)=x3+10 x2+65x+34.d,dn3.7,K46?2?lA?1,2 9fs2(x)=x3+10 x2+65x+34?n,A?1 2?-O4 3.L*?,12S(K46)?:9(1),1(9).q?=12,p=6,q=4,fs2(x)=(x+1)2(9 x).d,dn3.7,K46?2?l:9(1),1(9).n,?=0,p=6,q=4,K46?2?l?;?=1,DOI:10.12677/aam.2024.1341451553A?p=6,q=4,K46?2?ll?,?=12,p=6,q=4,K46?2?l?l?12?.3.4.KKln?2?l?!|?y?x?KKln?2?l.T(J2?z 4 KKln?(J,?2?KKln?l?l?(J.n3.10.-0,1)G KKln.oG?SA?1 9fs3(x)?o.,A?1?-2n4,f3(x)=x4+42nx3+2(l+5n+5ln9n2)+6l 6n+n2x2+2(5l 3l2+9n+16lnl2n17n2+ln2)+(4l+4l28ln)+43l l26n+2ln+2n2x+164(l2 ln l2n+ln2)+3(24l2+24ln+24l2n 24ln2)+2(4l+5l2+4n+3ln 9l2n 8n2+9ln2)+(4l+6l2 10ln 2l2n+2ln2)+1 2l 2l2 2n+3ln+l2n+n2 ln2.y.LG?:I,?2?l?S(G)Xe:Jn1In1J(n1)12J(n1)l3J(n1)(nl)J1(n1)0J1l2J1(nl)2Jl(n1)Jl1JlIlJl(nl)3J(nl)(n1)2J(nl)1J(nl)lJnlInl.S(G)+I2nk4 1,0 S(G)+I2n?A?-?n 4.X1S(G)?A?-?n 4.uS(G)?A?,d?Bs3?.?,Bs3=n 212l3(n l)n 10l2(n l)2(n 1)1l 1n l3(n 1)2ln l 1.dn2.6,Bs3?A?S(G)?A?.LO,Bs3?A?fs3(x).?n?y.3.11.?,kXe(J:(1)?=0,S(G)=A(G).d,KKln?2?lA?=?A?,z 4?n2.2(J.(2)?=12,S(G)=S(G).d,KKln?2?lA?=?lA?.(3)?=1,S(G)=D(G).d,KKln?2?lA?=lA?.3.12.KK46,X7.DOI:10.12677/aam.2024.1341451554A?Figure 7.KK46 7.KK46K?l?O:A(KK46)=011111000000101111000000110111000000111011000000111101000000111110111100000001011111000001101111000001110111000001111011000000111101000000111110,D(KK46)=011111222233101111222233110111222233111011222233111101222233111110111122222221011111222221101111222221110111222221111011333332111101333332111110,DOI:10.12677/aam.2024.1341451555A?oKK46?l?:S(KK46)=022222222233202222222233220222222233222022222233222202222233222220222222222222022222222222202222222222220222222222222022333332222202333332222220.LO,?A(KK46)?A?:fsA(x)=(x+1)8(x48x3+2x2+60 x+9),l?D(KK46)?A?:fSD(x)=(x+1)8(x4 8x3 176x2 246x 39),?l?S(KK46)?A?:fSAD(x)=(x+2)9(x3 18x2 134x 88).q?=0,n=6,l=4,fs3(x)=x4 8x3+2x2+60 x+9.d,dn3.10,KK46?2?lA?1 9fs3(x)=x4 8x3+2x2+60 x+9?o,A?1?-8.?/,?=1,n=6,l=4,fs3(x)=x4 8x3 176x2 246x 39.d,dn3.10,KK46?2?lA?1 9fs3(x)=x4 8x3 176x2 246x 39?o,A?1?-8.?=12,n=6,l=4,fs3(x)=12(x+1)(2x3 18x2 67x 22).d,dn3.10,KK46?2?lA?1 9gs(x)=2x3 18x2 67x 22?n,A?1?-9.-gs3(x)=x3 18x2 134x 88,Kgs3(2x)=4(2x3 18x2 67x 22).d,(S(KK46)?A?fSAD(x),12S(KK46)?A?1 9gs(x)=2x318x267x22?n,A?1?-9.n,?=0,n=6,l=4,KK46?2?l?;?=1,n=6,l=4,KK46?2?ll?,?=12,n=6,l=4,KK46?2?l?l?12?.4.(?,Hln,KqpKKln?2?l?1?x./?y,?9?AaA?2?l,=?,l,?l?1?2,?Y?l4KC?:.?e5?A(?,DOI:10.12677/aam.2024.1341451556A?3a?2?l4K,l?4l4?12.z1 Nikiforov,V.(2017)Merging the A-and Q-Spectral Theories.Applicable Analysis and DiscreteMathematics,11,81-107.https:/doi.org/10.2298/AADM1701081N2 Brondani,A.E.and Fran ca,F.A.M.and Oliveira,C.S.(2022)Positive Semidefiniteness ofA(G)on Some Families of Graphs.Discrete Applied Mathematics,323,113-123.https:/doi.org/10.1016/j.dam.2020.12.0073 Schultz,H.P.(1989)Topological Organic Chemistry.1.Graph Theory and Topological Indicesof Alkanes.Journal of Chemical Information and Computer Sciences,29,227-228.https:/doi.org/10.1021/ci00063a0124 Freitas,M.,Del-Vecchio,R.and Abreu,N.(2010)Spectral Properties of KKjnGraphs.Matem atica Contempor anea,39,129-134.https:/doi.org/10.21711/231766362010/rmc39155 Tait,M.and Tobin,J.(2017)Three Conjectures in Extremal Spectral Graph Theory.Journalof Combinatorial Theory.Series B,126,137-161.https:/doi.org/10.1016/j.jctb.2017.04.0066 Zhai,M.,Lin,H.and Wang,B.(2012)Sharp Upper Bounds on the Second Largest Eigenvaluesof Connected Graphs.Linear Algebra and Its Applications,437,236-241.https:/doi.org/10.1016/j.laa.2012.02.0047 Belardo,F.and Li Marzi,E.M.and Simi c,S.K.and Wang,J.(2010)On the Index of Necklaces.Graphs and Combinatorics,26,163-172.https:/doi.org/10.1007/s00373-010-0910-48 Brouwer,A.E.and Haemers,W.H.(2012)Spectra of Graphs.Springer,New York.https:/doi.org/10.1007/978-1-4614-1939-69 de Abreu,N.M.M.,Justel,C.M.,Markenzon,L.,Oliveira,C.S.and Waga,C.F.E.M.(2019)Block-Indifference Graphs:Characterization,Structural and Spectral Properties.Discrete Ap-plied Mathematics,269,60-67.https:/doi.org/10.1016/j.dam.2018.11.03410 Nath,M.and Paul,S.(2014)On the Distance Laplacian Spectra of Graphs.Linear Algebraand Its Applications,460,97-110.https:/doi.org/10.1016/j.laa.2014.07.02511 Rojo,O.(2011)Line Graph Eigenvalues and Line Energy of Caterpillars.Linear Algebra andIts Applications,435,2077-2086.https:/doi.org/10.1016/j.laa.2011.03.06412 Topcu,H.,Sorgun,S.and Haemers,W.H.(2016)On the Spectral Characterization of Pineap-ple Graphs.Linear Algebra and Its Applications,507,267-273.https:/doi.org/10.1016/j.laa.2016.06.01813 Pirzada,S.and Mushtaq U.(2023)On the Eigenvalues of the Distance Matrix of Families ofGraphs with Given Number of Pendent Vertices.Preprint.DOI:10.12677/aam.2024.1341451557A?