The Kn-complement of a graph G, denoted by Kn − G, is defined as the graph obtained from the complete graph Kn by removing a set of edges that span G; if G has n vertices, then Kn − G coincides with the complement G of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K n ± G, where K m n is the ...

We prove that a graph G contains no induced 5-vertex path and no induced complement of a 5-vertex path if and only if G is obtained from 5-cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement (where split unification is a new class-preserving operation that is introduced in this paper).

The Kn-complement of a graph G, denoted by Kn−G, is de ned as the graph obtained from the complete graph Kn by removing a set of edges that span G; if G has n vertices, then Kn − G coincides with the complement G of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K n ±G, where K n is the comple...

We determine the Castelnuovo-Mumford regularity of binomial edge ideals complement reducible graphs (cographs). For cographs with $n$ vertices maximum grows as $2n/3$. also bound by graph theoretic invariants and construct a family counterexamples to conjecture Hibi Matsuda.

Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of ...

the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...

a concept related to the spectrum of a graph is that of energy. the energy e(g) of a graph g is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of g . the laplacian energy of a graph g is equal to the sum of distances of the laplacian eigenvalues of g and the average degree d(g) of g. in this paper we introduce the concept of laplacian energy of fuzzy graphs. ...

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number λ(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complem...

In this paper we examine the classes of graphs whose Kn-complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of Kn, the Kn-complement of H is the graph Kn H which is obtained from Kn by removing the edges of H . Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of ...

For any graph G = (V, E), D  V is a global dominating set if D dominates both G and its complement G . The global domination number g(G) of a graph G is the fewest number of vertices required of a global dominating set. In general, max{(G), (G )} ≤ g(G) ≤ (G)+(G ), where (G) and (G ) are the respective domination numbers of G and G . We show, when G is a planar graph, that g(G) ≤ max{...