McCuaig and Ota proved that every 3-connected graph G on at least 9 vertices admits a contractible triple, i. e. a connected subgraph H on three vertices such that G − V (H) is 2-connected. Here we show that every 3-connected graph G on at least 9 vertices has more than |V (G)|/10 many contractible triples. If, moreover, G is cubic, then there are at least |V (G)|/3 many contractible triples, w...

Let G = (V,E) be a connected graph, An equitable dominating S of a graph G is called the neighborhood connected equitable dominating set (nced-set) if the induced subgraph 〈Ne(S)〉 is connected The minimum cardinality of a nced-set of G is called the neighborhood connected equitable domination number of G and is denoted by γnce(G). In this paper we initiate a study of this parameter. For any gra...

Let G be a 2-connected graph. A subset D of V (G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove that γ2(G) ≤ n ln δ(G) δ(G) (1 + oδ(1)) where oδ(1) denotes a function that tends to 0 as δ →...

Zhang, F. and X. Guo, Reducible chains in several types of 2-connected graphs, Discrete Mathematics 105 (1992) 285-291. Let F& 4, $ and 8 denote the sets of all 2-connected graphs, minimally 2-connected graphs, critically 2-connected graphs, and critically and minimally 2-connected graphs, respectively. We introduce the concept of %,-reducible chains of a graph G in %,, i = 0, 1, 2, 3, and give...

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. A graph G is called rainbow k-connected, if there is an edge-colouring of G with k colours such that G is rainbow-connected. In this talk we will study rainbow k-connected graphs with a minimum number of edges. For an integer n ≥ 3 and 1 ≤ k ≤ n− 1 let t(n, k) denote the ...

For a connected graph G = (V,E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P . A path P is called a monophonic path if it is a chordless path. A longest x− y monophonic path is called an x− y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path, for some x an...

Let $G$ be a connected graph on $n$ vertices. $G$ is called tricyclic if it has $n + 2$ edges, and tetracyclic if $G$ has exactly $n + 3$ edges. Suppose $mathcal{C}_n$ and $mathcal{D}_n$ denote the set of all tricyclic and tetracyclic $n-$vertex graphs, respectively. The aim of this paper is to calculate the minimum and maximum of eccentric connectivity index in $mathcal{C}_n$ and $mathcal{D}_n...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

A graph G is k-Hamilton-connected (k-hamiltonian) if G−X is Hamilton-connected (hamiltonian) for every setX ⊂ V (G) with |X| = k. In the paper, we prove that (i) every 5-connected claw-free graph with minimum degree at least 6 is 1Hamilton-connected, (ii) every 4-connected claw-free hourglass-free graph is 1-Hamilton-connected. As a byproduct, we also show that every 5-connected line graph with...