Norm. distance Laplacian & verification of examples (or lack thereof)

### Defining the matrices def distLapla(G): n=G.order() M=zero_matrix(QQ,n,n) for i in range(n): for j in range(i+1,n): x=G.distance(i,j) M[i,j]=-x M[j,i]=-x M[i,i]=M[i,i]+x M[j,j]=M[j,j]+x return M def InvTransMatrix(G): n=G.order() M=zero_matrix(QQ,n,n) for i in range(n): for j in range(i+1,n): x=G.distance(i,j) M[i,j]=0 M[j,i]=0 M[i,i]=M[i,i]+x M[j,j]=M[j,j]+x for i in range(n): M[i,i]=M[i,i]^(-1) return M def InvTransDistLapla(G): M=InvTransMatrix(G)*distLapla(G) return M 

# Definition of transmission sequence def trs(G): n=G.order() ones = ones_matrix(n,1) dg=G.distance_matrix() trg=vector(dg*ones) trgs=trg.list() trgs.sort() return trgs # Checks if a graph is transmission regular def IsTrsReg(G): unique_list_t=[] trans=list(trs(G)) for y in trans: if y not in unique_list_t: unique_list_t.append(y) if len(unique_list_t)==1: return True else: return False 

### Cospectral graphs found in paper: C. Reinhart. The Normalized Distance Laplacian. Special Matrices, 9 (2021) 1-18. ### Copied to lists at the end of this code 

##### THE NUMBER OF CONNECTED COMPONENTS OF THE COMPLEMENT ####### 

### Finds cospectral graphs with different number of connected components of the complement, no results on 8 for x in L8: unique_list = [] components=[] for y in x: Gbar=Graph(y).complement() components.append(Gbar.connected_components_number()) for y in components: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

### Finds cospectral graphs with different number of connected components of the complement for x in L9: unique_list = [] components=[] for y in x: Gbar=Graph(y).complement() components.append(Gbar.connected_components_number()) for y in components: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

['H~uj~Uz', 'H~vj|~j']
[3, 4]

['H~uj~Uz', 'H~vj|~j']
[3, 4]

### Finds cospectral graphs with different number of connected components of the complement for x in L10: unique_list = [] components=[] for y in x: Gbar=Graph(y).complement() components.append(Gbar.connected_components_number()) for y in components: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

['I~vz|~n}w', 'IbCIHEHh?']
[5, 1]

['I~vz|~n}w', 'IbCIHEHh?']
[5, 1]

A1=Graph('IbCIHEHh?') A2=Graph('I~vz|~n}w') 

A1 

P=graphs.PetersenGraph() P.is_isomorphic(A1) 

True

True

P 

P.complement() 

A2 

A1.complement() 

(A1.complement()).connected_components_number() 

1

1

A2.complement() 

(A2.complement()).connected_components_number() 

5

5

InvTransDistLapla(A1).charpoly()-InvTransDistLapla(A2).charpoly() 

0

0

InvTransDistLapla(A1).charpoly() 

x^10 - 10*x^9 + 222/5*x^8 - 2872/25*x^7 + 23861/125*x^6 -
660126/3125*x^5 + 486504/3125*x^4 - 230256/3125*x^3 + 63504/3125*x^2 -
7776/3125*x

x^10 - 10*x^9 + 222/5*x^8 - 2872/25*x^7 + 23861/125*x^6 - 660126/3125*x^5 + 486504/3125*x^4 - 230256/3125*x^3 + 63504/3125*x^2 - 7776/3125*x

#### DIAMETER AND TRANSMISSION REGULARITY (evidence of no example on <=10 vertices) 

### Finds cospectral graphs with different diameter, no results on 8 for x in L8: unique_list = [] diameter=[] for y in x: diameter.append(Graph(y).diameter()) for y in diameter: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

### Finds cospectral graphs with different diameter, no results on 9 for x in L9: unique_list = [] diameter=[] for y in x: diameter.append(Graph(y).diameter()) for y in diameter: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

### Finds cospectral graphs with different diameter, no results on 10 for x in L10: unique_list = [] diameter=[] for y in x: diameter.append(Graph(y).diameter()) for y in diameter: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

### Finds cospectral graphs with different transmission sequence, no results on 8 for x in L8: unique_list=[] transseq=[] for y in x: transseq.append(trs(Graph(y))) for y in transseq: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: print x print unique_list 

### Finds cospectral graphs with different transmission sequence for x in L9: unique_list=[] transseq=[] for y in x: transseq.append(trs(Graph(y))) for y in transseq: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: for y in x: if IsTrsReg(Graph(y))==True: print x print unique_list 

### Finds cospectral graphs with different transmission sequence for x in L10: unique_list=[] transseq=[] for y in x: transseq.append(trs(Graph(y))) for y in transseq: if y not in unique_list: unique_list.append(y) if len(unique_list)>1: for y in x: if IsTrsReg(Graph(y))==True: print x print unique_list 

['I~vz|~n}w', 'IbCIHEHh?']
[[10, 10, 10, 10, 10, 10, 10, 10, 10, 10], [15, 15, 15, 15, 15, 15, 15,
15, 15, 15]]
['I~vz|~n}w', 'IbCIHEHh?']
[[10, 10, 10, 10, 10, 10, 10, 10, 10, 10], [15, 15, 15, 15, 15, 15, 15,
15, 15, 15]]

['I~vz|~n}w', 'IbCIHEHh?']
[[10, 10, 10, 10, 10, 10, 10, 10, 10, 10], [15, 15, 15, 15, 15, 15, 15, 15, 15, 15]]
['I~vz|~n}w', 'IbCIHEHh?']
[[10, 10, 10, 10, 10, 10, 10, 10, 10, 10], [15, 15, 15, 15, 15, 15, 15, 15, 15, 15]]

#### VERIFICATION THAT THE PATH GRAPH IS SPECTRALLY DETERMINED UP TO 10 VERTICES 

P10=graphs.PathGraph(10) for x in L8: for y in x: if Graph(y).is_isomorphic(P10)==True: print x 

for x in L9: for y in x: if Graph(y).is_isomorphic(P10)==True: print x 

for x in L10: for y in x: if Graph(y).is_isomorphic(P10)==True: print x 

#### COSPECTRAL GRAPHS ON 10 AND FEWER VERTICES 

L8=[['GfC}@o', 'GraK[S']] L8 

[['GfC}@o', 'GraK[S']]

[['GfC}@o', 'GraK[S']]

L9=[['HnVyKEB', 'HnS}kEB'],['HxEwk^_', 'HnTyKEB'],['H~uj~Uz', 'H~vj|~j'],['HnVykEB', 'HnU}kEB']] L9 

[['HnVyKEB', 'HnS}kEB'],
 ['HxEwk^_', 'HnTyKEB'],
 ['H~uj~Uz', 'H~vj|~j'],
 ['HnVykEB', 'HnU}kEB']]

[['HnVyKEB', 'HnS}kEB'],
 ['HxEwk^_', 'HnTyKEB'],
 ['H~uj~Uz', 'H~vj|~j'],
 ['HnVykEB', 'HnU}kEB']]

L10=[['In`Mh~ii_', 'IzJilMXYg'], ['IrXMLEMM_', 'IvCj[UHIW'], ['IbdiLeIiG', 'IbDiLmIiG'], ['IbNiLeJ}G', 'IvWYhULhw'], ['IzOKHM|ko', 'IjOKLM|ko'], ['IrDJi]G{W', 'IrDJg}G{W'], ['IvUIh]igW', 'IbW{zEYwW'], ['IrAmhM]wG', 'Ivh\\hEI]?'], ['Ib@WjElwO', 'IjCi]eiGG'], ['IraZZfHN_', 'IbDz]ewiW'], ['IjzYlMWYG', 'IbWMtMdno'], ['Ij]YlEMlo', 'IzgyWmJZW'], ['IjQjxfWhO', 'IjoH}^WhO'], ['If_yXuGwO', 'Ij@[|U?mG'], ['IjyghMywo', 'Itsgx^BZG'], ['Iv?^HVGio', 'IbOJpMdlo'], ['Iz?hhekwO', 'Ib?Hh]G^w'], ['IbQHzfYjO', 'IboNpMl|?'], ['Iv]Kh}kgW', 'IzqixMIww'], ['IbC{HvHgo', 'IfHYkEJIo'], ['Ib`iH~Y}?', 'Ijd[nEiiG'], ['IvYwheMgO', 'IjGJImiz_'], ['IfQhxFWyO', 'IfqGhnIXW'], ['IrQyxfWhO', 'IrQmxfWhO'], ['IbCM\\eWiO', 'If@GlM{gO'], ['IrTWXEIiW', 'IbPh[NWhO'], ['IjIIlMWYG', 'IfIIlMWYG'], ['InHyLEjyO', 'IfHYLFnyO'], ['IfCHLEiiO', 'IbPKgM[iG'], ['Inc}I}MiO', 'IfVKp}fi_'], ['I~DGwMGgW', 'IzDIwMGgW'], ['IbowjMYh?', 'IbDJMEwyO'], ['IzPKhMZlW', 'IbvilmiWG'], ['IbOLLNMI_', 'IfHYWmGkO'], ['Ib@ih]iyG', 'Ij@Ih]ijG'], ['In[]JNMjW', 'IjW]t}d|G'], ['IfkKxmGxo', 'IbSGrf{yo'], ['IjYwxFY{o', 'IfiwNUZMg'], ['Ij`Xlvij_', 'Ij`Wlvijg'], ['IrJIje]i?', 'If_Xh^gjO'], ['IzOGhMzlO', 'I~CmKehIW'], ['IjQkX^Wgo', 'IrdZh]GgW'], ['IvCm{EHKO', 'IrKi\\UHKO'], ['IrPJhuMmG', 'IrL]YEXwW'], ['IbtiNemi_', 'IzUiLeJJW'], ['IvYGzM]wO', 'IjDL]NwjO'], ['IfcjLuHKO', 'Ibkn{EHKO'], ['IbsKnMKlO', 'IfUKjUKlO'], ['Ijsj^EJJO', 'IvSn\\UHIW'], ['IrThgMGiW', 'IbhYKMxKO'], ['IrFWh~Jm_', 'IvyINEyN_'], ['If_wXuGxO', 'IrGmgm\\g?'], ['IbRXhVWiO', 'IjU[JEJN?'], ['IfPiHu}jw', 'IfdmXumy_'], ['IfIg\\eNyW', 'IvSxWM\\Iw'], ['Ir`WXFI}?', 'Ir`WXFI^?'], ['IbC[jNHz?', 'IbQKpm{i_'], ['IbfX}Uwho', 'Ir^iYeiho'], ['IfCLzNGZO', 'IfCL^NGNO'], ['IzDhMUmIO', 'IjFgImYhW'], ['IbPiXmyyO', 'IvFKyMHYW'], ['IbhWjFJiO', 'IvC\\HNHYG'], ['IrELJFHN?', 'IbFLJFHN?'], ['IrELJFHN?', 'IbELjFHN?'], ['IbFLJFHN?', 'IbELjFHN?'], ['Ifk\\L~gYG', 'IbiXJ~\\M_'], ['IvJGJuZNo', 'IjNL[uwYo'], ['Ivhg\\mG^G', 'IfFHLmwjW'], ['IjDKl]i}?', 'IjY[jEYhW'], ['IfKI\\mijG', 'IrVKwMJ[W'], ['IfCw]mGhG', 'IbCg]}GjG'], ['IbumjVkM?', 'IjXIKnxn?'], ['IraZZfHNo', 'IbDz]e{iW'], ['IbVnHUIgO', 'IfKJKmyi?'], ['InTk~E|hO', 'Ibygzneyo'], ['IvW]jVLhW', 'IvW]jULhw'], ['Ird[LNGYW', 'IrY[ZfGIo'], ['IfPj^ejWG', 'IbZLZFy]?'], ['IjuMHVhgW', 'Ir\\jgMGjW'], ['IvTMrmcj_', 'IrtMrmcj_'], ['IjDLmUwiO', 'IrFJ[FgNO'], ['Ijt[lUiiG', 'IrW^HNxLO'], ['IziYGMJWw', 'IrPW{uHxG'], ['IbFJ[FiMO', 'IbFHMVhMO'], ['IzRkXNYgo', 'In?}x]{WG'], ['Ib[\\ZFi{W', 'Ir|NlNHGW'], ['IbYgxFYwo', 'IbELZUw}?'], ['IvPKrmcj_', 'IfOLznchG'], ['I~E{jEMiO', 'Ifc\\LnXYG'], ['IjOH[M{xO', 'IrK]YEXgW'], ['IfEGjMMmO', 'Ib@iHmyyG'], ['IjXHnUJi?', 'IjDgInZkO'], ['IbgInEmyO', 'IfKmHmNi?'], ['IvL\\h^Ni?', 'Ijf|ZvGgo'], ['Irgg~uYh?', 'Iv\\hgMGjW'], ['IjQ\\xVGjO', 'IrjJLEJ]G'], ['IbAHlU\\]?', 'IbCLHNXZO'], ['IfkMHm^i?', 'IzC{KEmmW'], ['IrL]GEiLW', 'Ijck^e?ZG'], ['IbPHLE^N?', 'IjSgYNIkO'], ['IrTWXEYmW', 'IbPh[NWzO'], ['IbQ^^fWgO', 'IlSH|FZN?'], ['IbiJNEM^?', 'IbAMjV[NO'], ['IbjMG}ymO', 'IdsijfLXo'], ['IrEZXeMi_', 'IfVLZUgM?'], ['IrUyhMIgW', 'IrUIhMygW'], ['IjOkjeGhO', 'IbO{jeGhO'], ['IzDGymHgW', 'IzDGyNHgW'], ['IbW[WFG[o', 'IjCi]EiGG'], ['Irvgjeigw', 'Ibvgjmigw'], ['In@i\\eLWG', 'IbHIGm{}O'], ['IrC\\L~HYG', 'IfDzMEwmO'], ['IbRWxe\\yO', 'IrDmHNYMg'], ['IbD^hNWZW', 'InoJ{MWzO'], ['IbMiJMGkO', 'IvgGjFIWO'], ['IbgH|mG|?', 'Ifgjg}GhO'], ['Ifdm\\emy_', 'IzUjLmJJW'], ['Ijdi|eZj_', 'IrW^LNxNO'], ['IrCMXexi?', 'IjQ]\\EkGG'], ['IbNyHEHhw', 'IbNYHEHjw'], ['IbxWjFJiO', 'IvC\\HNhYG'], ['InZK|EXgO', 'IrUkzVKM?'], ['IrLi}MXgW', 'IvWYi]hhW'], ['IfC[LNLnG', 'IbC[LnLnG'], ['IbpwjMYh?', 'IjDJMEwyO'], ['IjUxhVWiO', 'IjR[zEJN?'], ['IbqyNFg]o', 'IrhY^eHXW'], ['IrTGiMHkW', 'IrTGgmHwW'], ['IfIWxeMYO', 'IrAmZekM?'], ['IrdZlMGMw', 'IjCyLMZxo'], ['Ir?}\\VHXW', 'IbSgvNklG'], ['Ir\\HgMGhW', 'IbhYKelKO'], ['Ib@GzeYiG', 'IrGygEHkW'], ['Ivogjm}h?', 'IfQGzvWzO'], ['IbQ\\ZfWgO', 'IlSHlFJN?'], ['IbPHLE^NG', 'IjTgYNIkO'], ['IrLi}MWiW', 'IvW]g]hhW'], ['IfBGlM|kO', 'IzVmpMEg?'], ['IjsghMihO', 'IbOMpMdnO'], ['Ij}YlMWYG', 'IbWNtMdn_'], ['IbGMMm{~_', 'I~LIYNWgw'], ['Iv_\\HnXYG', 'Ifc\\LnHYG'], ['IzDkwMHhw', 'IzDlwMHhW'], ['IjQg|VWgO', 'IloG|FBNO'], ['Ivj^G]X[W', 'Irj^G]\\[W'], ['IbPHLEZNG', 'IjDgYNIkO'], ['IvxZHMXYW', 'IrdJk^g^O'], ['IfJGjeIiG', 'IbiXJFHM_'], ['IbNGjeIiG', 'IbaXJFHm_'], ['IneihUHgW', 'IbWmxMWwW'], ['IrIJgm\\g?', 'IrG}GmZg?'], ['IvoivMll_', 'Ifoivmll_'], ['I~Q\\ZMPMw', 'IzR\\ZMPmg'], ['IjR]jEJN?', 'I~DjMEwiO'], ['IbQL~FWhO', 'IfGIhmhzO'], ['IjfmHUHGW', 'IjMiMENgO'], ['Ir\\WH^jxo', 'Izk[Ln]JO'], ['IbcyjeMi?', 'IbOk^NWgo'], ['IjwWhMmgO', 'Ib@M\\EhnO'], ['IfYg~FWgO', 'IlRIlFBNO'], ['Ir\\ilnXIW', 'IvE}[UXZg'], ['Ijgi|eKwO', 'IngWxeKwo'], ['IrK{hVLgo', 'InGILm{z?'], ['IfJKjeIiG', 'IbmXJFHM_'], ['IvyGzM]wO', 'IjDN]NwjO'], ['IbNyHEHhW', 'IbVJJFHgW'], ['Ib_XhVIhW', 'IbQXZNWgO'], ['IzdyIUHko', 'IjS|]EwhG'], ['IbI[jeM~_', 'If@mx^G\\g'], ['InJHjmiiG', 'IzUJLMlM_'], ['Ib_XhVMhW', 'IbQXZnWgO'], ['InowlUMgO', 'IrU\\ZFHM?'], ['IbAHlu^]?', 'Ibgjwmhj?'], ['IvIilMWYG', 'IziIlMWYG'], ['IvTJgMGgW', 'IrxYKEhKO'], ['IjOkLN[go', 'IbAHjVL\\_'], ['IvFg|MHWW', 'IfRKpmui_'], ['Ifgg|EI[_', 'IfAGnELNO'], ['IrVHkmhWw', 'IjFH]Uwlo'], ['IzDkwMHgw', 'IzDLwMHhW'], ['IfXIl^i}?', 'InFi}UhWW'], ['IbIIlMWYg', 'IbIIlMWYW'], ['IrlzI^X[W', 'Id}mhFfZo'], ['IfIIheij_', 'IrZZPERKG'], ['IfWGlEigO', 'IjOMHEwiO'], ['IfcyLmIy_', 'IrtKwmHwW'], ['I~YMWMZLW', 'IrLi~M[gW'], ['IjDLKFgMO', 'IjEgYEIhW'], ['IrOwjNlmO', 'IfIwJmjm_'], ['Ijc~HVGko', 'IfC{H^Lio'], ['IbAGnEkMO', 'IbQyREBKG'], ['IrPmpmEg?', 'IrPmpMFg?'], ['IrUXjVGiO', 'IrK]YeXgW'], ['IbOIxe\\yO', 'IjAkJNHmG'], ['Ib~GhMZ{O', 'IrVHkMimW'], ['IboJhUHwW', 'In`W[]?Yg'], ['IzmYhuHgW', 'IzvGYMWhw'], ['IbkNJEGgO', 'IbkNHeGgO'], ['IvqghmyWG', 'IjDgYnZkO'], ['IzQWjULmO', 'IrQWjVLmO'], ['IrYWheMgO', 'IjGIImij_'], ['InWLn^GI?', 'IjLg]^eGG'], ['Ib?MJNKMW', 'Iz`kPsDBG'], ['IjOYjVkho', 'IrFHJ}Hm_'], ['IbQYjVlgO', 'Ijkgl]IGw'], ['Ijsi\\EJJO', 'IvSj[UHIW'], ['IbhwjFjiO', 'IzdLwMHhW'], ['IbKIM}zj?', 'IbtH{eyHo'], ['IbMIHnIg?', 'IbEh]fGGG'], ['InHWlUIig', 'IrW^GNXLO'], ['Ib[\\JFIhW', 'IbW{jEYwW'], ['IvCiKeHKO', 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['IbW]WFG]o', 'IjCj]FiGG'], ['IrYJgm\\g?', 'IrW}GmZg?'], ['IboHzUZwW', 'IvEh]nkGG'], ['IfQHzfWhO', 'IvGIkmhz?'], ['IrWm^mI}?', 'I~AkJMZNW'], ['IfOZHvGiO', 'IbPIpMtl?'], ['InQXxEyWG', 'IvCl]ViGG'], ['IbVNJFHN?', 'IbmLZFHN?'], ['IrPkHNWgo', 'IfAkX^gWG'], ['Ibdi^emi_', 'IzUiKmJJW'], ['In`Ih~ii_', 'IrJilMXYg'], ['IbJIHfJi?', 'IfsghFBX?'], ['IfHw\\VYgO', 'IvDwXVJgO'], ['Ijh[nMzn_', 'InU^J]yhW'], ['IvkXhVHhO', 'IzQ^XVWgO'], ['IvoWLNK[W', 'IvpGLM[[W'], ['IbPkNEmMO', 'IbGYKnwz?'], ['Ijc^HvWgO', 'IjSi\\EJMW'], ['IzPKhM\\ko', 'IrdZlMGYW'], ['IrdmHNWMg', 'Irdjg}GwW'], ['IbTGzeI}G', 'IrI}gENgW'], ['IfI~[E\\MW', 'Ibm\\HfXiw'], ['IfWIlFwio', 'IrAXjFJ|?'], ['IrTWXEY{W', 'IbOh]NWzO'], ['IbDLKFg]W', 'IjegYEIiW'], ['IrDyLeYig', 'IzgY[mHXW'], ['IjwWhMMgO', 'Ib@M\\EhNO'], ['Ib^XlUIiO', 'IbB{LNYMo'], ['InLi{UxWW', 'Ivogrmdn_'], ['IvuIh^IkW', 'IbFh\\vh[W'], ['IrUGh]yhO', 'IjS\\[MWxO'], ['IjvH{uxHo', 'IzVg[fJ]W'], ['Ibu^NeH]_', 'IfFHLM|mo'], ['IbOmJNWgO', 'InPWXUIWG'], ['IjPyjEGwo', 'IzJhtceAG'], 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