Complete Graph

The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see Paley). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their automorphism groups may be much larger than the groups of the corresponding schemes. We determine the parameters for which the graphs are connected, or equivalently, the schemes are primitive. Also we prove that generalised Paley graphs are sometimes isomorphic to Hamming graphs and consequently have large automorphism groups, and we determine precisely the parameters for this to occur. We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in LLP to distinguish between cyclotomic schemes and s...

A set S of vertices in a graph G(V, E) is called a dominating set if every vertex v ∈ V is either an element of S or is adjacent to an element of S. A set S of vertices in a graph G(V, E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. The domination number of a graph G denoted by γ(G) is the minimum cardinality of a dominating set in G. Respectively the total domination number of a graph G denoted by γt(G) is the minimum cardinality of a total dominating set in G. An upper bound for γt(G) which has been achieved by Cockayne and et al. in [1] is: for any graph G with no isolated vertex and maximum degree Δ(G) and n vertices, γt(G) ≤ n − Δ(G )+1 . Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for γt(G). Also, for circular complete graphs, we determine the value of γt(G).

This paper is designed to bring about results related to the chromatic polynomials of graphs. Specifically, this paper demonstrates various ways or techniques in determining the chromatic polynomials of some special graphs such as barbell graph, book graph, gear graph, helm graph, ladder graph and sun graph. As a highlight of this work, the chromatic polynomials of the Kr -gluing of the complete graphs and corona of graphs are
obtained. In the generation of these results, earlier relevant works of various authors are used.

In this paper, we deal with the notion of star coloring of graphs. A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored. We give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as hypercubes, tori, d-dimensional grids, graphs with bounded treewidth and planar graphs.

Let be a 2-factorization of the complete graph Kv admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex-set V(Kv) can then be identified with the point-set of AG(n, p) and each 2-factor of is the union of p-cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2-(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs

One of the most useful measures of cluster quality is the modularity of a partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random (unstructured) graph. In this paper we show that the problem of finding a partition maximizing the modularity of a given graph G can be reduced to a minimum weighted cut problem on a complete graph with the same vertices as G. We then show that the resulted minimum cut problem can be efficiently solved with existing software for graph partitioning and that our algorithm finds clusterings of a better quality and much faster than the existing clustering algorithms.

The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For integer $k\geq 1$, the $k$-tuple total domination number $\gamma_{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$, which is a set of vertices in $G$ such that every vertex of $G$ is adjacent to at least $k$ vertices in it. For existing this number, must the minimum degree of $G$ is at least $k$. Here, we study the $k$-tuple total domination number in inflated graphs when $k\geq 2$. First we prove that $n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1$, and then we characterize graphs $G$ that the $k$-tuple total domination number number of $G_I$ is $n(...

A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv∈ E(G) and |deg(u) − deg(v)| ≤ 1. The minimum cardinality of such a dominating set is denoted by í µí»¾ í µí±’ (í µí°º) and is called equitable domination number of G. In this paper we introduce the Equitable domination to the cross product of special graph.

Let H1;H2;:::;Hk+1 be a sequence of k + 1 nite, undirected, simple graphs. The (mul- ticolored) Ramsey number r(H1;H2;:::;Hk+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k + 1 colors, there is a monochromatic copy of Hi in color i for some 1 i k + 1. We describe a general technique that supplies tight lower bounds for several numbers r(H1;H2;:::;Hk+1) when k 2, and the last graph Hk+1 is the complete graph Km on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K3;K3;Km) = ( m3poly logm), thus solving (in a strong form) a conjecture of Erd} os and S os raised in 1979. Another special case of our result implies that r(C4;C4;Km) = ( m2poly logm) and that r(C4;C4;C4;Km) = ( m2= log 2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain esti- mates of character sums.

Global, relative, and local complexity of the five Platonic solids (tetrahedron, octahedron, cube, icosahedron, and dodecahedron) are described and compared. Several of the most recent measures of topological complexity are used: the subgraph count, overall connectivity and overall Wiener indices, the total walk count, and the information theoretic index for vertex de- grees distribution. Equations are derived for the first several orders of these indices as func- tions of the number of vertices and vertex degrees. Relative complexity, defined as the ratio of the complexity index selected and its value for the complete graph having the same number of vertices as the respective Platonic solid, singles out tetrahedron as the most complex structure with 100 % relative complexity. The global complexity indices, as well as the local indices (de- fined per vertex and per edge) uniformly identify icosahedron as the most complex Platonic solid. These findings correlate with the preferable f...

We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs.