Degree sequence

BenE74 Bender, E. A., Canfield, E. R. The asymptotic number of labelled graphs with given degree sequence. J. Combinatorial Theory Ser. A 24 (1978) 296-307. 

HanP79 Hanlon, P. Enumeration of graphs by degree sequence. J. Graph Theory 3 (1979) 295-299. 

From the distribution of the element values in the i th row of the detour matrix, the maximum path degree sequence of the i th vertex is derived as a local vector-descriptor defined as ... 

From the frequencies of the row entries, the vertex distance code (or distance degree sequence of a vertex) is defined as the ordered sequence of the occurrences of the increasing distance values for the i th considered vertex,... 

Quintas, L.V. and Slater, P.J. (1981). Pairs of Non-Isomorphic Graphs Having the Same Path Degree Sequence. MATCH (Comm.Math.Comp.Chem.), 12, 75-86. 

Randic et al.301 studied the complexity of lower fullerenes by using the distance degree sequences DDSs for all symmetry non-equivalent vertices, the multiplicity of the DDSs, which are defined as the cardinality of each equivalence class, the augmented degrees for all vertices of different equivalence class ( (see text above),292 294 which represent a measure of the local... 

The counting polynomial is a description of a graph property in terms of a sequence of numbers, such as the distance degree sequence or the sequence of the number of k independent edge sets [Hosoya, 1988,1990 Trinajstic, 1992 Diudea, Gutman et al., 2001 Noy, 2003 Diudea, Vizitiu et al., 2007]. The counting polynomial is defined as... 

For instance, the distance degree sequence ofvertex 2 is DDS2 = 3, 1, 1, which means that there are three vertices one bond away from V2, one vertex located at distance two from V2, and one vertex at distance three. The graph distance code is GDC = 5, 5, 3, 2 and the graph distance index is CDI —25 + 25 + 9 + 4 = 63. 

The last item shows that the characterization of bond degree sequences of molecules does not refer to the dominemce order. This important partial order was mentioned since we shall use it later in cormection with chiral permutational isomers. 

Remark (Degree sequences of graphs by cyclomatic number) A partition a h 2m of an even number... 

I. Faradzhev. Generation of nonisomorphic graphs with a given degree sequence. n Algorithmic Studies in Combinatorics, pages 11-19. NAUKA, Moscow, 1978. 

M. Schocker. On degree sequences of graphs with given cyclomatic number. Publ. Inst Math. (Beograd) (N.S.), 69(83) 34-40,2001. 

However, some models for random geometric graphs exist in the literature which produce a predefined power-law distribution of the degree sequence of the vertices. 

Sparse matrices The augmented valence, the walks of length two, and the distance degree sequence matrix... 

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