Graphs with a Given Degree Sequence

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Aug 18, 2006 - the size of a spanning forest, its component order sequence, its girth and the number of girth cycles present, and its canonical edge list.
Graphs with a Given Degree Sequence Peter Adams∗†

Roger B. Eggleton‡

James A. MacDougall§ August 18, 2006

Abstract The level set G(n, m) comprises all unlabelled simple graphs of order n and size m, and is partitioned into similarity classes, comprising all graphs with the same degree sequence. When graphs are ordered lexicographically by their signature, a unique numerical list of structural descriptors, the similarity classes of G(n, m) occur in contiguous blocks; the first graph in each similarity class is its sentinel. The sentinel of the first similarity class in each G(n, m) is determined, and shown to be the unique realization of its degree sequence. The degree sequence of the last similarity class in each G(n, m) is also determined, as are the exact size range for which it has more than one realization, and the exact size range for which its sentinel has more than one component. 2000 Mathematics subject classification: primary 05C07, secondary 05C70, 05C75.

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Degree Sequences and signatures

We shall discuss degree sequences of graphs (unlabelled finite simple graphs). Let us call two graphs similar if they have the same degree sequence. Similarity is an equivalence relation, so the similarity classes of graphs of order n form a partition of G(n), the set of all graphs of order n. If the graph G ∈ G(n) has degree sequence d, we call each graph in the similarity class [G] a realization of d, and we call |[G]| the multiplicity of d. Our main purpose here is to specify a canonical realization of each degree sequence, ∗ Research

supported by the Australian Research Council of Physical Sciences, Univ. of Queensland, QLD 4072, Australia ‡ Mathematics Department, Illinois State Univ., Normal, IL 61790-4520, USA § School of Mathematical and Physical Sciences, The Univ. of Newcastle, NSW 2308, Australia † School

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which we shall call the sentinel of its similarity class, and to study some sentinels of particular interest. Typically a degree sequence has many realizations. For example (see [3]), G(10) comprises 12005168 graphs, but only 16016 similarity classes, so the mean multiplicity of a graphic sequence of order 10 is nearly 750. Indeed, the two largest similarity classes, corresponding to the complementary degree sequences 2334445566 and 3344555667, each have multiplicity 65500. At the other extreme, 2252 similarity classes are singletons: their degree sequences are uniquely graphic. So that we can assign a linear order to all graphs, with each graph G we associate a unique numerical sequence Σ(G), the signature of G. For a given G, the signature Σ(G) specifies in turn its order, its size, its degree sequence, the size of a spanning forest, its component order sequence, its girth and the number of girth cycles present, and its canonical edge list. (The edge list by itself uniquely specifies G, but is preceded in the signature by other parameters of structural interest.) For instance, we shall later encounter a family of graphs that includes the graph G with signature Σ(G) = 5 | 8 | 23344 | 4 5 | 3 5 | 12 13 14 15 23 24 25 34. The vertical bars help parse the signature to make it more readily readable. It specifies that G has order 5, size 8, and degree sequence 23344; any spanning forest is a tree of size 4, and its one component has order 5; the girth (smallest cycle size) of G is 3, and G contains 5 cycles of size 3; and finally, for all possible assignments of [1..5] as vertex labels, the lexicographically earliest resulting edge list is 12, 13, 14, 15, 23, 24, 25, 34. It is easily verified that G is produced when two adjacent edges are deleted from the complete graph of order 5, so G = K5 − E(K1,2 ). We shall also encounter a family of graphs that includes the graph H with signature Σ(H) = 5 | 8 | 33334 | 4 5 | 3 4 | 12 13 14 15 23 24 35 45. In this case H = K5 − E(2K2 ), the graph produced by deleting two independent edges from K5 . We sort the signatures of all finite graphs into lexicographic order, defined so that Σ(G) < Σ(H) holds for any graphs G and H essentially when the first place s in which their signatures differ satisfies s(G) < s(H). This yields a unique linear ordering s(H). This exception is for the historical reason that when all preceding parameters for G and H are equal, Steinbach chose to list G before H if G has more girth cycles than H. Quite apart from the fact that the canonical edge sequence of a graph G fully determines G, hence determines all the parameters preceding it

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in Σ(G), there is further redundancy within the segment of Σ(G) preceding the canonical edge sequence. For example, given the size m and the truncated degree sequence d1 ≤ d2 ≤ · · · ≤ dn−1 we could deduce the order n from the length of this sequence, and the maximum degree dn = 2m − Σi