The degree sequences of self-complementary graphs (original) (raw)
Self-complementary graphs and generalisations: A comprehensive reference manual
University of Malta, 1999
A graph which is isomorphic to its complement is said to be a self-complementary graph, or sc-graph for short. These graphs have a high degree of structure, and yet they are far from trivial. Suffice to say that the problem of recognising self-complementary graphs, and the problem of checking two sc-graphs for isomorphism, are both equivalent to the graph isomorphism problem.
A graph and its complement with specified properties. IV. Counting self-complementary blocks
Journal of Graph Theory, 1981
In this series, we investigate the conditions under which both a graph G and its complement G possess certain specified properties. We now characterize all the graphs G such that both G and G have the same number of endpoints, and find that this number can only be 0 or 1 or 2. As a consequence, we are able to enumerate the self-complementary blocks. 1. NOTATIONS AND BACKGROUND In the first paper [l] in this series, we found all graphs G such that both G and its complement have connectivity 1, and other properties. In the second paper [2], we determined the graphs G for which G and G are obtained from some graph by the same unary operation. More recently [ 3 ] we characterized the graphs such that both G and C? have the same girth and the same circumference 3 or 4. An endpoint of graph has degree 1. We denote the number of endpoints in G by e = e (G) and in G by 2. We characterize all the graphs G with e = 2 (? 2) in the next section, and count the number of selfcomplementary blocks in the last section.
D Equienergetic Self Complementary Graphs - 2009
The D-eigenvalues {µ 1 , µ 2 , . . . , µ n } of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by spec D (G) . The D-energy E D (G) of the graph G is the sum of the absolute values of its D-eigenvalues. We describe here the distance spectrum of some self-complementary graphs in the terms of their adjacency spectrum. These results are used to show that there exists D-equienergetic self-complementary graphs of order n = 48t and 24(2t + 1) for t ≥ 4 .
Procedia Computer Science, 2015
A finite sequence d: d 1 , d 2 , d 3 ,. .. . , d n of nonnegative integers is said to be graphical if there exists some finite simple graph G, having vertex set V={v 1 , v 2 , v 3 , …. , v n } such that each v i has degree d i (1 ≤ i ≤ n). In this paper we have proposed an algorithm that takes a non-increasing sequence as input and determines whether the given degree sequence is graphic by constructing the adjacency matrix from the given sequence in non-increasing order and checking whether it can be orthogonally diagonalizable. The matrix generated in this process can be used to determine a lot of interesting information regarding the characteristic of the graph directly from the given degree sequence.
On almost self-complementary graphs
Discrete Mathematics, 2006
A graph is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. Almost self-complementary circulant graphs were first studied by Dobson andŠajna in 2004. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost selfcomplementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.
A graph and its complement with specified properties V: The self‐complement index
Mathematika, 1980
The self-complement index s(G) of a graph G is the maximum order of an induced subgraph of G whose complement is also induced in G. This new graphical invariant provides a measure of how close a given graph is to being selfcomplementary. We establish the existence of graphs G of order p having s(G) = n for all positive integers n < p. We determine s(G) for several families of graphs and find in particular that when G is a tree, s{G) = 4 unless G is a star for which s(G) = 2. §1. The self-complement index and the induced number. Our purpose is to propose invariants which, in some sense, measure the degree to which a graph is selfcomplementary. To this end we define two related invariants which satisfy this requirement. We then show that the two are equivalent. Throughout we use the notation and terminology of [2]. In particular, all graphs are finite, without loops or multiple lines. The order of a graph G is the number p of points in it. And if X is a set of points in a graph G then we use <.Y> to denote the subgraph of G induced by X. The self-complement index of a graph G, denoted s(G), is defined as the order of the largest induced subgraph H of G, such that H is also induced in G. For a graph G of order p it is clear that 1 < s(G) < p as we do not include the null graph in the family of graphs; see Figure 1 in [3]. Now a related invariant of a graph G is defined. The induced number m(G) is the minimum order of a graph which contains both G and G as induced subgraphs. The first result indicates that s(G) and m(G) are essentially identical. We then proceed to show that s(G) partitions the graphs of order p into p classes which are nonempty except when s(G) = p and p = 2 or 3 (mod 4). The number s(G) is then derived for several important families of graphs. §2. The equivalence of the two invariants. THEOREM 1. If G is a graph of order p with self-complement index s(G) and induced number m{G), then m{G) = 2p-s(G). Proof. Let s = s(G), m = w(G), and let H of order s be a largest induced subgraph of G whose complement is also induced in G. To prove the upper bound, we construct a graph F of order 2p-s in which both G and G are induced subgraphs. Consider disjoint copies of G and G. Let U c V(G)
More on almost self-complementary graphs
Discrete Mathematics, 2009
A graph X is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. If I is a perfect matching in X c and ϕ : X → X c − I is an isomorphism, then the graph X is said to be fairly almost self-complementary if ϕ preserves I setwise, and unfairly almost self-complementary if it does not. In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary. Two perfect matchings I and J in X c are said to be X-non-isomorphic if no isomorphism from X + I to X + J induces an automorphism of X. We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X-nonisomorphic perfect matchings for every even order greater than or equal to four.
Constructing bounded degree graphs with prescribed degree and neighbor degree sequences
Discrete Applied Mathematics
Let D = d 1 , d 2 ,. .. , d n and F = f 1 , f 2 ,. .. , f n be two sequences of positive integers. We consider the following decision problems: is there a i) multigraph, ii) loopless multigraph, iii) simple graph, iv) connected simple graph, v) tree, vi) caterpillar G = (V, E) such that for all k, d(v k) = d k and w∈N (v k) d(w) = f k (d(v) is the degree of v and N (v) is the set of neighbors of v). Here we show that all these decision problems can be solved in polynomial time if max k d k is bounded. The problem is motivated by NMR spectroscopy of hydrocarbons.
Solution of the Hamiltonian problem for self-complementary graphs
Journal of Combinatorial Theory, Series B, 1979
A graph G is said to be highly constricted if there exists a nonempty subset S of vertices such that (i) G-S has more than I S I components, (ii) S induces the complete graph, and (iii) for every I(E S and v # S, we have do(u) > do(v), where do(u) denotes the degree of u in G. In this paper it is shown that a non-hamiltonian self-complementary graph G of order p is highly constricted, unless p = 4N and G is a particular graph G*(4N). It is also proved that if G is a se&complementary graph of order p(>8) and = its degree sequence, then G is pancyclic if r has a realization with a hamiltonian cycle, and G has a 2-factor if * has a realization with a 2-factor, unless p = 4N and G = G*(4N).