The dual of the category of generalized trees.
The Dynamics of the Forest Graph Operator
In 1966, Cummins introduced the “tree graph”: the tree graph T(G) of a graph G (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge, i.e., two spanning trees T1 and T2 are adjacent if T2 = T1 − e + f for some edges e ∈ T1 and f ∉ T1. The tree graph of a connected graph need not be connected. To obviate this difficulty we define the “forest graph”: let G be a labeled graph of order α, finite or infinite,...
The eavesdropping number of a graph
Let be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices and , a minimum -separating set is a smallest set of edges in whose removal disconnects and . The edge connectivity of , denoted , is defined to be the minimum cardinality of a minimum -separating set as and range over all pairs of distinct vertices in . We introduce and investigate the eavesdropping number, denoted , which is defined to be the maximum cardinality of...
The Eccentric Connectivity Polynomial of some Graph Operations
The eccentric connectivity index of a graph G, ξ^C, was proposed by Sharma, Goswami and Madan. It is defined as ξ^C(G) = ∑ u ∈ V(G) degG(u)εG(u), where degG(u) denotes the degree of the vertex x in G and εG(u) = Max{d(u, x) | x ∈ V (G)}. The eccentric connectivity polynomial is a polynomial version of this topological index. In this paper, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented.* The work...
The edge C₄ graph of some graph classes
The edge C₄ graph of a graph G, E₄(G) is a graph whose vertices are the edges of G and two vertices in E₄(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C₄. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C₄ graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E₄(G). It is shown that for any graph G without isolated vertices, there...
The edge domination problem
An edge dominating set of a graph is a set D of edges such that every edge not in D is adjacent to at least one edge in D. In this paper we present a linear time algorithm for finding a minimum edge dominating set of a block graph.
The edge geodetic number and Cartesian product of graphs
For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several...
The edge-count criterion for graphic lists.
The edge-minimal polyhedral maps of Euler characteristic .
The Edmonds-Gallai decomposition for the -piece packing problem.
The enumeration of bit-sequences that satisfy local criteria.
The Estrada Index
The Estrada Index: An Updated Survey
The Euler characteristic of a category.
The evolution of uniform random planar graphs.
The excessive [3]-index of all graphs.
The existence of 2-factors in squares of graphs
The Existence of Quasi Regular and Bi-Regular Self-Complementary 3-Uniform Hypergraphs
A k-uniform hypergraph H = (V ;E) is called self-complementary if there is a permutation σ : V → V , called a complementing permutation, such that for every k-subset e of V , e ∈ E if and only if σ(e) ∉ E. In other words, H is isomorphic with H′ = (V ; V(k) − E). In this paper we define a bi-regular hypergraph and prove that there exists a bi-regular self-complementary 3-uniform hypergraph on n vertices if and only if n is congruent to 0 or 2 modulo 4. We also prove that there exists a quasi regular...
The extremal connectivity of the strictly weak digraph
The extremal irregularity of connected graphs with given number of pendant vertices
The irregularity of a graph is defined as the sum of imbalances over all edges , where denotes the degree of the vertex in . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with vertices and pendant vertices (), and characterize the corresponding extremal graphs.