The bipartite vertex (resp. edge) frustration of a graph G, denoted by ψ(G) (resp. φ(G)), is the smallest number of vertices (resp. edges) that have to be deleted from G to obtain a bipartite subgraph of G. A sharp lower bound of the bipartite vertex frustration of the line graph L(G) of every graph G is given. In addition, the exact value of ψ(L(G)) is calculated when G is a forest.

In 1972, M. Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere S in such a way that two vertices joined by an edge have distance more than √ 3 (i.e. distance more than 2π/3 on the sphere). In 1978, D. Larman [4] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed p 8/3. In addition, he conjectured ...

Preface Graphs are often used as a model for networks, for example transport networks, telecommunication systems, a network of servers and so on. Typically, the networks and thus the representing graphs are connected; that means, that there exists a path between all pairs of two vertices in the graph. But sometimes an element of the network fails, for the graphs that means the removal of a vert...

For a connected graph G = (V , E), an edge set S ⊂ E is a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum krestricted edge cut. For U1,U2 ⊂ V (G), denote the set of edges of Gwith one end in U1 and the other in U2 by [U1,U2]. Define ξk(G...

A block in a graph G is a maximal subgraph with no cut-vertex. Blocks are either 2-connected graphs, or the graph with two vertices and only one edge, or, in the case where G is just a single vertex, G itself. Definition 1.1. Let C be a class of connected graphs, and B ⊆ C the class of 2-connected graphs in C (B may contain the graph formed by an edge). We call C stable if it satisfies the foll...

For a subset S of edges in a connected graph G, the set S is a k-restricted edge cut if G− S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. A connected graph G is said to be λk-connected if G has a k-restricted edge cut. Let ξk(G) = min{|[X, X̄ ]| : |...

let g_1 and g_2 be simple connected graphs with disjoint vertex sets v(g_1) and v(g_2), respectively. for given vertices a_1in v(g_1) and a_2in v(g_2), a splice of g_1 and g_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the :union: of g_1 and g_2. in this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

A digraph is said to be super-connected if every minimum vertex cut is the out-neighbor set or in-neighbor set of a vertex. A digraph is said to be reducible, if there are two vertices with the same out-neighbor set or the same in-neighbor set. In this paper, we prove that a strongly connected arc-transitive oriented graph is either reducible or super-connected. Furthermore, if this digraph is ...

‎let $g$ be a graph with vertex set $v(g)$ and edge set $x(g)$ and consider the set $a={0,1}$‎. ‎a mapping $l:v(g)longrightarrow a$ is called binary vertex labeling of $g$ and $l(v)$ is called the label of the vertex $v$ under $l$‎. ‎in this paper we introduce a new kind of graph energy for the binary labeled graph‎, ‎the labeled graph energy $e_{l}(g)$‎. ‎it depends on the underlying graph $g$...

Let K be an abelian group and G be a connected graph, both finite. Using basic properties of circulations, we show that it is easy to generate uniformly random K-circulations on G. This leads to efficient algorithms for computing the cut edges, cut edge-pairs, and cut vertices of a graph; for example, the cut edges are “usually” the edges where a random circulation vanishes. In the distributed ...