Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k, L)-complex. The two conditions are symmetry properties of the graph, namely star-transitivity and st(edge)-transitivity. In this paper we investigate star-transitive and st(edge)-transitive graphs by studying the structure of the vertex and edge stabilisers of suc...

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdős-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an ...

A connected graph is double connected if its complement is also connected The following Ramsey type theorem is proved in this paper There exists a function h n de ned on the set of integers exceeding three such that every double connected graph on at least h n vertices must contain as an induced subgraph a special double connected graph which is either one of the following graphs or the complem...

A connected graph is double-connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), deened on the set of integers exceeding three, such that every double-connected graph on at least h(n) vertices must contain, as an induced subgraph, a special double-connected graph, which is either one of the following graphs or th...

Acharya and Hegde have introduced the notion of strongly k-indexable graphs: A (p, q)-graph G is said to be strongly k-indexable if its vertices can be assigned distinct integers 0, 1, 2, ..., p − 1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices can be arranged as an arithmetic progression k, k+ 1, k + 2, ..., k + (q − 1). Such an assignment ...

Let G be a simple graph on n vertices having edge-connectivity 1(' (G) > O. We say G is k-critical if 1(' (G) :::: k and 1(1 (G-e) < k for every edge e of G. We denote by ~(n,k) the set of all k-critical graphs on n vertices. In this paper we prove that the maximum number of edges of a graph G in ~(n,k) to be: k(n-k) if n ~ 3k; and L ~ (n+k)2 J. if k + 1 s n < 3k. Further, we characterise the e...

An edge e of a simple 3-connected graph G is essential if neither the deletion G\e nor the contraction G/e is both simple and 3–connected. Tutte’s Wheels Theorem proves that the only simple 3–connected graphs with no non-essential edges are the wheels. In earlier work, as a corollary of a matroid result, the authors determined all simple 3-connected graphs with at most two non-essential edges. ...

‎let g=(v,e) be a simple connected graph with vertex set v and edge set e. the first, second and third zagreb indices of g are respectivly defined by: $m_1(g)=sum_{uin v} d(u)^2, hspace {.1 cm} m_2(g)=sum_{uvin e} d(u).d(v)$ and $ m_3(g)=sum_{uvin e}| d(u)-d(v)| $ , where d(u) is the degree of vertex u in g and uv is an edge of g connecting the vertices u and v. recently, the first and second m...