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Degree Sequence of Graph Operator for some Standard Graphs

Harisha
Harisha
, 
P. S Ranjini
P. S Ranjini
, 
V. Lokesha
V. Lokesha
 et 
Sandeep Kumar
Sandeep Kumar
   | 10 août 2020
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Publié en ligne: 10 août 2020
Pages: 99 - 108
Reçu: 10 avr. 2019
Accepté: 05 juin 2019
DOI: https://doi.org/10.2478/amns.2020.2.00018
Mots clés
© 2020 Harisha et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Graphs with the same DS
Graphs with the same DS

Fig. 2

P4,C6,S7,K6,K3,4,R – regular
P4,C6,S7,K6,K3,4,R – regular

Fig. 3

Titania nanotube TiO2[m,n] with degree assigned to the vertices
Titania nanotube TiO2[m,n] with degree assigned to the vertices

Degree Sequence of vertex-R join graph for path, complete graph, cycle, star, complete bipartite and r–regular graphs

G1G2DS(G1∨˙RG2)DS({G_1}{\dot \vee _R}{G_2})
PxPy{(2 + y)2, (4 + y)x−2, 2x−1, (1 + x)2, (2 + x)y−2}
PxKy{(2 + y)2, (4 + y)x−2, 2x−1, (x + y − 1)y}
PxCy{(2 + y)2, (4 + y)x−2, 2x−1, (2 + x)y}
PxSy{(2 + y)2, (4 + y)x−2, 2x−1, (1 + x)y−1, (x + y − 1)}
PxK(y, z){(2 + y + z)2, (4 + y + z)x−2, 2x−1, (y + x)z, (z + x)y}
Pxr – regular{(2 + y)2, (4 + y)x−2, 2x−1, (r + x)y}
KxPy{(2x − 2 + y)x, 2x(x−1)/2, (1 + x)2, (2 + x)y−2}
KxKy{(2x − 2 + y)x, 2x(x−1)/2, (x + y − 1)y}
KxCy{(2x − 2 + y)x, 2x(x−1)/2, (2 + x)y}
KxSy{(2x − 2 + y)x, 2x(x−1)/2, (1 + x)y−1, (y − 1 + x)}
KxK(y, z){(2x − 2 + y + z)x, 2x(x−1)/2, (x + y)z, (x + z)y}
Kxr – regular{(2x − 2 + y)x, 2x(x−1)/2, (r + x)y}
CxPy{(4 + y)x, 2x, (1 + x)2, (2 + x)y−2}
CxKy{(4 + y)x, 2x, (y − 1 + x)}
CxCy{(4 + y)x, 2x, (2 + x)y}
CxSy{(4 + y)x, 2x, (1 + x)y−1, (y − 1 + x)}
CxK(y, z){(4 + y + z)x, 2x, (y + x)z, (z + x)y}
Cxr – regular{(4 + y)x, 2x, (r + x)y}
G1G2DS(G1∨˙RG2)DS({G_1}{\dot \vee _R}{G_2})
SxPy{(2 + y)x−1, (2x − 2 + y), 2x−1, (1 + x)2, (2 + x)y−2}
SxKy{(2 + y)x−1, (2x − 2 + y), 2x−1, (y − 1 + x)y}
SxCy{(2 + y)x−1, (2x − 2 + y), 2x−1, (2 + x)y}
SxSy{(2 + y)x−1, (2x − 2 + y), 2x−1, (1 + x)y−1, (y − 1 + x)}
SxK(y, z){(2 + y + z)x−1, (2x − 2 + y + z), 2x−1, (z + x)y, (y + x)z}
Sxr – regular{(2 + y)x−1, (2x − 2 + y), 2x−1, (r + x)y}
Kx,yPz{(2x + z)y, (2y + z)x, 2xy, (1 + x + y)2, (2 + x + y)z−2}
Kx,yKz{(2x + z)y, (2y + z)x, 2xy, (z − 1 + x + y)z}
Kx,yCz{(2x + z)y, (2y + z)x, 2xy, (2 + x + y)z}
Kx,ySz{(2x + z)y, (2y + z)x, 2xy, (1 + x + y)z−1, (z − 1 + x + y)}
Kx,yKz,t{(2x + z + t)y, (2y + z + t)x, 2xy, (x + y + z)t, (x + y + t)z}
Kx,yr – regular{(2x + z)y, (2y + z)x, 2xy, (r + x + y)z}
r – regularPy{(2r + y)x, 2rx/2, (1 + x)2, (2 + x)y−2}
r – regularKy{(2r + y)x, 2rx/2, (y − 1 + x)y}
r – regularCy{(2r + y)x, 2rx/2, (2 + x)y}
r – regularSy{(2r + y)x, 2rx/2, (y − 1 + x), (1 + x)y}
r – regularK(y, z){(2r + y + z)x, 2rx/2, (y + x)z, (z + x)y}
r1 – regularr2 – regular{(2r1 + y)x, 2r1x/2, (r2 + x)y}

Degree sequence of subdivision for path, complete graph, cycle, star, complete bipartite and r–regular graphs

G1G2G1∨˙SG2{G_1}{\dot \vee _S}{G_2}
PxPy{(1 + y)2, (2 + y)x−2, 2x−1, (1 + x)2, (2 + x)y−2}
PxKy{(1 + y)2, (2 + y)x−2, 2x−1, (y − 1 + x)y}
PxCy{(1 + y)2, (2 + y)x−2, 2x−1, (2 + x)y}
PxSy{(1 + y)2, (2 + y)x−2, 2x−1, (1 + x)y−1, (y − 1 + x)}
PxK(y, z){(1 + y + z)2, (2 + y + z)x−2, 2x−1, (y + x)z, (z + x)y}
Pxr – regular{(1 + y)2, (2 + y)x−2, 2x−1, (r + x)y}
KxPy{(x − 1 + y)x, 2x(x−1)/2, (1 + x)2, (2 + x)y−2}
KxKy{(x − 1 + y)(2x), 2x(x−1)/2}
KxCy{(x − 1 + y)x, 2x(x−1)/2, (2 + x)y}
KxSy{(x − 1 + y)x, 2x(x−1)/2, (1 + x)y−1, (y − 1 + x)}
KxK(y, z){(x − 1 + y + z)x, 2x(x−1)/2, (x + y)z, (x + z)y}
KxCxr – regularPy{(x − 1 + y)x, 2x(x−1)/2, (r + x)y}{(2 + y)x, 2x, (1 + x)2, (2 + x)y−2}
CxKy{(2 + y)x, 2x, (y − 1 + x)y}
CxCy{(2 + y)x, 2x, (2 + x)y}
CxSy{(2 + y)x, 2x, (1 + x)y−1, (y − 1 + x)}
CxK(y, z){(2 + y)x, 2x, (y + x)z, (z + x)y}
Cxr – regular{(2 + y)x, 2x, (r + x)y}
SxPy{(1 + y)x−1, (x − 1 + y), 2x−1, (2 + x)y−2}
SxKy{(1 + y)x−1, (x − 1 + y), 2x−1, (y − 1 + x)y}
SxCy{(1 + y)x−1, (x − 1 + y), 2x−1, (2 + x)y}
SxSy{(1 + y)x−1, (x − 1 + y), 2x−1, (1 + x)y−1, (y − 1 + x)}
G1G2G1∨˙SG2{G_1}{\dot \vee _S}{G_2}
SxK(y, z){(1 + y + z)x−1, (x − 1 + y + z), 2x−1, (z + x)y, (y + x)z}
Sxr – regular{(1 + y)x−1, (x − 1 + y), 2x−1, (r + x)y}
Kx,yPz{(x + z)y, (y + z)x, 2xy, (1 + x + y)2(2 + x + y)z−2}
Kx,yKz{(x + z)y, (y + z)x, 2xy, (1 + x + y) (z − 1 + x + y)z}
Kx,yCz{(x + z)y, (y + z)x, 2xy, (2 + x + y)z}
Kx,ySz{(x + z)y, (y + z)x, 2xy, (1 + x + y)z−1, (z − 1 + x + y)}
Kx,yKz,t{(x + z + t)y, (y + z + t)x, 2xy, (x + y + z)t, (x + y + t)z}
Kx,yr – regular{(x + z)y, (y + z)x, 2xy, (r + x + y)z}
r – regularPy{(r + y)x, 2rx/2, (1 + x)2, (2 + x)x−2}
r – regularKy{(r + y)x, 2rx/2, (y − 1 + x)y}
r – regularCy{(r + y)x, 2rx/2, (2 + x)y}
r – regularSy{(r + y)x, 2rx/2, (y − 1 + x), (1 + x)y−1}
r – regularK(y,z){(r + y + z)x, 2rx/2, (y + x)z, (z + x)y}
r1 – regularr2 – regular{(r1 + y)x, 2r1x/2, (r2 + x)y}

j.amns.2020.2.00018.tab.001.w2aab3b7d849b1b6b1ab1b1c23Aa

GVerticesEdgesDegree Sequence
Pnn(n − 1){12, 2n−2}
Knnn(n − 1)/2{(n − 1)n}
Cnnn{2n}
Snn(n − 1){1n−1, (n − 1)1}
K(m,n)m + nmn{mn, nm}
r – regularn(n − 1){rn}

Degree sequence of vertex-T join graph for path, complete graph, cycle, star, complete bipartite and r–regular graphs.

G1G2DS(G1∨˙TG2)DS({G_1}{\dot \vee _T}{G_2})
PxPy{(2 + y)2, (4 + y)x−2, 32, 4(x − 3), (1 + x)2, (2 + x)y−2}
PxKy{(2 + y)2, (4 + y)x−2, 32, 4(x − 3), (y − 1 + x)y}
PxCy{(2 + y)2, (4 + y)x−2, 32, 4x−3, (2 + x)y}
PxSy{(2 + y)2, (4 + y)x−2, 32, 4x−3, (1 + x)y−1, (x + y − 1)}
PxK(y, z){(2 + y + z)2, (4 + y + z)x−2, 32, 4x−3, (y + x)z, (z + x)y}
Pxr – regular{(2 + y)2, (4 + y)x−2, 32, 4x−3, (r + x)y}
KxPy{(2x − 2 + y)x, (2x − 2)x(x−1)/2, (1 + x)2, (2 + x)y−2}
KxKy{(2x − 2 + y)x, (2x − 2)x(x−1)/2, (x + y − 1)y}
KxCy{(2x − 2 + y)x, (2x − 2)x(x−1)/2, (2 + x)y}
KxSy{(2x − 2 + y)x, (2x − 2)x(x−1)/2, (1 + x)y−1, (y − 1 + x)}
KxK(y, z){(2x − 2 + y + z)x, (2x − 2)x(x−1)/2, (x + y)z, (x + z)y}
Kxr – regular{(2x − 2 + y)x, (2x − 2)x(x−1)/2, (r + x)y}
CxPy{(4 + y)x, 4x, (1 + x)2, (2 + x)y−2}
CxKy{(4 + y)x, 4x, (y − 1 + x)y}
CxCy{(4 + y)x, 4x, (2 + x)y}
CxSy{(4 + y)x, 4x, (1 + x)y−1, (y − 1 + x)}
CxK(y, z){(4 + y + z)x, 4x, (y + x)z, (z + x)y}
Cxr – regular{(4 + y)x, 4x, (r + x)y}
SxPy{(2 + y)x−1, (2x − 2 + y), xx−1, (1 + x)2, (2 + x)y−2}
SxKy{(2 + y)x−1, (2x − 2 + y), xx−1, (y − 1 + x)y}
SxCy{(2 + y)x−1, (2x − 2 + y), xx−1, (2 + x)y}
SxSy{(2 + y)x−1, (2x − 2 + y), xx−1, (1 + x)y−1, (y − 1 + x)}
SxK(y, z){(2 + y + z)x−1, (2x − 2 + y + z), xx−1, (z + x)y, (y + x)z}
Sxr – regular{(2 + y)x−1, (2x − 2 + y), xx−1, (r + x)y}
G1G2DS(G1∨˙TG2)DS({G_1}{\dot \vee _T}{G_2})
Kx,yPz{(2x + z)y, (2y + z)x, (x + y)xy, (1 + x + y)2, (2 + x + y)z−2}
Kx,yKz{(2x + z)y, (2y + z)x, (x + y)xy, (z − 1 + x + y)z}
Kx,yCz{2x + z)y, (2y + z)x, (x + y)xy, (2 + x + y)z}
Kx,ySz{(2x + z)y, (2y + z)x, (x + y)xy, (1 + x + y)z−1, (z − 1 + x + y)}
Kx,yKz,t{(2x + z + t)y, (2y + z + t)x, (x + y)xy, (x + y + z)t, (x + y + t)z}
Kx,yr – regular(2x + z)y, (2y + z)x, (x + y)xy, (r + x + y)z
r – regularPz{(2r + y)x, (2r)xr/2, (1 + x)2, (2 + x)y−2}
r – regularKy{(2r + y)x, (2r)xr/2, (x + y − 1)y}
r – regularCy{(2r + y)x, (2r)xr/2, (2 + x)y}
r – regularSy{(2r + y)x, (2r)xr/2, (1 + x)y−1, (x + y − 1)}
r – regularKy,z{(2r + y + z)x, (2r)xr/2, (x + y)z, (x + z)y}
r1 – regularr2 – regular{(2r1 + y)x, (2r1)xr1/2, (r − 2 + x)y}

Degree Sequence of vertex-Q join graph for path, complete graph, cycle, star, complete bipartite and r–regular graphs

G1G2DS(G1∨˙QG2)DS({G_1}{\dot \vee _Q}{G_2})
PxPy{(1 + y)2, (2 + y)x−2, 32, 4(x − 3)2, (1 + x)2, (2 + x)y−2}
PxKy{(1 + y)2, (2 + y)x−2, 32, 4(x − 3), (y − 1 + x)y}
PxCy{(1 + y)2, (2 + y)x−2, 32, 4x−3, (2 + x)y}
PxSy{(1 + y)2, (2 + y)x−2, 32, 4x−3, (1 + x)y−1, (x + y − 1)}
G1G2DS(G1∨˙QG2)DS({G_1}{\dot \vee _Q}{G_2})
PxK(y, z){(1 + y + z)2, (2 + y + z)x−2, 32, 4x−3, (y + x)z, (z + x)y}
Pxr – regular{(1 + y)2, (2 + y)x−2, 32, 4x−3, (r + x)y}
KxPy{(x − 1 + y)x, (2x − 2)x(x−1)/2, (1 + x)2, (2 + x)y−2}
KxKy{(x − 1 + y)x, (2x − 2)x(x−1)/2, (x + y − 1)y}
KxCy{(x − 1 + y)x, (2x − 2)x(x−1)/2, (2 + x)y}
KxSy{(x − 1 + y)x, (2x − 2)x(x−1)/2, (1 + x)y−1, (y − 1 + x)}
KxK(y, z){(x − 1 + y + z)x, (2x − 2)x(x−1)/2, (x + y)z, (x + z)y}
Kxr – regular{(x − 1 + y)x, (2x − 2)x(x−1)/2, (r + x)y}
CxPy{(2 + y)x, 4x, (1 + x)2, (2 + x)y−2}
CxKy{(2 + y)x, 4x, (y − 1 + x)y}
CxCy{(2 + y)x, 4x, (2 + x)y}
CxSy{(2 + y)x, 4x, (1 + x)y−1, (y − 1 + x)}
CxK(y, z){(2 + y + z)x, 4x, (y + x)z, (z + x)y}
Cxr – regular{(2 + y)x, 4x, (r + x)y}
SxPy{(1 + y)x−1, (x − 1 + y), xx−1, (1 + x)2, (2 + x)y−2}
SxKy{(1 + y)x−1, (x − 1 + y), xx−1, (y − 1 + x)y}
SxCy{(1 + y)x−1, (x − 1 + y), xx−1, (2 + x)y}
SxSy{(1 + y)x−1, (x − 1 + y), xx−1, (1 + x)y−1, (y − 1 + x)}
SxK(y, z){(1 + y + z)x−1, (x − 1 + y + z), xx−1, (z + x)y, (y + x)z}
Sxr – regular{(1 + y)x−1, (x − 1 + y), xx−1, (r + x)y}
Kx,yPz{(x + z)y, (y + z)x, (x + y)xy, (1 + x + y)2, (2 + x + y)z−2}
Kx,yKz{(x + z)y, (y + z)x, (x + y)xy, (z − 1 + x + y)z}
Kx,yCz{(x + z)y, (y + z)x, (x + y)xy, (2 + x + y)z}
Kx,ySz{(x + z)y, (y + z)x, (x + y)xy, (1 + x + y)z−1, (z − 1 + x + y)}
Kx,yKz,t{(x + z + t)y, (y + z + t)x, (x + y)xy, (x + y + z)t, (x + y + t)z}
Kx,yr – regular{(x + z)y, (y + z)x, (x + y)xy, (r + x + y)z}
r – regularPz{(r + y)x, (2r)xr/2, (1 + x)2, (2 + x)y−2}
r – regularKy{(r + y)x, (2r)xr/2, (x + y − 1)y}
r – regularCy{(r + y)x, (2r)xr/2, (2 + x)y}
r – regularSy{(r + y)x, (2r)xr/2, (1 + x)y−1, (x + y − 1)}
r – regularKy,z{(r + y + z)x, (2r)xr/2, (x + y)z, (x + z)y}
r1 – regularr2 – regular{(r1 + y)x, (2r1)xr1/2, (r − 2 + x)y}
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