A graph G is collapsible if for every even subset X ~ V(G), G has a subgraph r such that G E(J ') is connected and the set of odd-degree vertices of J' is X. A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G. In this paper. we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between t...

Linguists often represent the relationships between words in a collection of text as an undirected graph G = (V,E), were V is the vocabulary and vertices are adjacent in G if and only if the words they represent co-occur in a relevant pattern in the text. Ideally, the words with similar meanings give rise to the vertices of a component of the graph. However, many words have several distinct mea...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E), called the underlying graph of S and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}, called the signature of S. The ×-line sigraph of S denoted by L×(S) is a sigraph defined on the line graph L(S u) of the graph Su by assigning to each edge ef of L(Su), the product of si...

topological graph (AT-graph): A = (G, R); G = (V, E) is a graph, R ⊆ ( E 2 ) in a topological graph T ... RT = set of crossing pairs of edges AT-graph A is realizable if there exists a topological graph T which is a drawing of G and RT = R. simply realizable ... T is simple rectilinearly realizable ... edges of T are straight-line segments weakly realizable ... RT ⊆ R Abstract topological graph...

In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph.

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from the line multigraphs of cubic multigraphs by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every e...

For integers k > 0 and r > 0, a conditional (k, r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex v of degree d(v) in G is adjacent to vertices with at least min{r, d(v)} different colors. The smallest integer k for which a graph G has a conditional (k, r)-coloring is called the rth order conditional chromatic number, denoted by χr(G). For different val...

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most bn/2c new edges...

The aim of this work is to investigate the F-index and F-coindex of the line graphs of the cycle graphs, star graphs, tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts. F-index of the line graph of subdivision graph of square grid graph, 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] are also investigated here.