A study on the curling number of graph classes

2007

We study integer sequences associated with the cyclic graph C r and the complete graph K r . Fourier techniques are used to characterize the sequences that count walks of length n on both these families of graphs. In the case of the cyclic graph, we show that these sequences are associated with an induced colouring of Pascal's triangle. This extends previous results concerning the Jacobsthal numbers.

Israel Journal of Mathematics, 1970

Necessary and sufficient conditions for a sequence (Pl, P2 .... p.) of positive integers to be the power sequence of a connected graph on n vertices with m edges are given. The maximum power of a connected graph on n vertices with m edges and the class of all extremal graphs are also determined.

2007

We study integer sequences associated to the cyclic graph C_r and the complete graph K_r. Fourier techniques are used to characterise the sequences that count walks of length n on both these families of graphs. In the case of the cyclic graph, we show that these sequences are associated to an induced colouring of Pascal's triangle. This extends previous results concerning the Jacobsthal numbers.

2021

A finite non-increasing sequence of positive integers 3 = (31 > · · · > 3=) is called a degree sequence if there is a graph = (+, ) with + = {E1, . . . , E=} and deg(E8) = 38 for 8 = 1, . . . , =. In that case we say that the graph realizes the degree sequence 3. We show that the exact number of labeled graphs that realize a fixed degree sequence satisfies a simple recurrence relation. Using this relation, we then obtain a recursive algorithm for the exact count. We also show that in the case of regular graphs the complexity of our algorithm is better than the complexity of the same enumeration that uses generating functions.

For a graph G whose degree sequence is d 1 , . . . , d n ,

Discrete Applied Mathematics, 2007

We enumerate weighted graphs with a certain upper bound condition. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that if the given graph is a bipartite graph, then its generating function is of the form p(x) (1 x)m+1 , where m is the number

Acta Universitatis Sapientiae, Informatica, 2013

In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (G z (n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is G z (30) = 5 876 236 938 019 300 and on n = 31 vertices is G z (31) = 22 974 847 474 172 374. Due to Corollary 21 in these results give the number of degree sequences of simple graphs on 30 and 31 vertices.

Acta Universitatis Sapientiae, Mathematica

Let ℕ0 be the set of all non-negative integers and 𝒫(ℕ0) be its power set. Then, an integer additive set-indexer (IASI) of a given graph G is an injective function f : V(G) → P(ℕ0) such that the induced function f+ : E(G) → 𝒫(ℕ0) defined by f+(uv) = f(u) + f(v) is also injective. An IASI f is said to be a weak IASI if |f+(uv)| = max(|f(u)|, |f(v)|) for all u, v ∈ V(G). A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph G, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph G is the minimum number of edges with singleton set-labels, required for a graph G to admit a weak IASI. In this paper, we study the admissibility of weak IASI by certain graph powers and their corresponding sparing numbers.

Mathematics, 2015

Let N 0 be the set of all non-negative integers and P(N 0 ) be its power set. Then, an integer additive set-indexer (IASI) of a given graph G is defined as an injective function f : V (G) → P(N 0 ) such that the induced edge-function f + :

For any graph class G and any two positive integers i and j, the Ramsey number RG(i, j) is the smallest positive integer such that every graph in G on at least RG(i, j) vertices has a clique of size i or an independent set of size j. For the class of all graphs, Ramsey numbers are notoriously hard to determine, and the exact values are known only for very small values of i and j. For planar graphs, a formula for determining all Ramsey numbers was obtained by Steinberg and Tovey (J. Comb. Theory Ser. B 59 (1993), 288-296). On the other hand, it is highly unlikely that such a formula will ever be found for claw-free graphs, as there exist infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Inspired by this contrast between known results on planar graphs and claw-free graphs, we initiate a systematic study of Ramsey numbers for graph classes. We give exact formulas for determining all Ramsey numbers for the class of perfect graphs and many of its subclasses. We also establish all Ramsey numbers for line graphs and long circular interval graphs, identified by Chudnovsky and Seymour as "the two principal basic classes of claw-free graphs", as well as for fuzzy circular interval graphs, another important subclass of clawfree graphs that contains all long circular interval graphs. As complementary results, we determine all Ramsey numbers for circular-arc graphs and cactus graphs.