Extending Hall's theorem into list colorings: a partial history.
Extending the dual of the Petersen graph.
Extending the MAX Algorithm for Maximum Independent Set
The maximum independent set problem is an NP-hard problem. In this paper, we consider Algorithm MAX, which is a polynomial time algorithm for finding a maximal independent set in a graph G. We present a set of forbidden induced subgraphs such that Algorithm MAX always results in finding a maximum independent set of G. We also describe two modifications of Algorithm MAX and sets of forbidden induced subgraphs for the new algorithms.
Extension of several sufficient conditions for Hamiltonian graphs
Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph , 1 ≤ r ≤ s ≤ t or the graph . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity of a graph...
Extension of strongly regular graphs.
Extension of -labelings of quadratic graphs.
Extensions of Foster's averaging formula to infinite networks with moderate growth.
Extensions of -subsets to -subsets - existence versus constructability
Extremal algebraic connectivities of certain caterpillar classes and symmetric caterpillars.
Extremal bipartite graphs with a unique k-factor
Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size...
Extremal Crossing Numbers of Complete -Chromatic Graphs
Extremal graph theory for metric dimension and diameter.
Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph
In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic...
Extremal Matching Energy of Complements of Trees
Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have...
Extremal problems for forbidden pairs that imply hamiltonicity
Let C denote the claw , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does...
Extremal problems for -partite and -colorable hypergraphs.
Extremal Properties of the Chromatic Polynomials of Connected 3-chromatic Graphs
Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants and
I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values...
Extremal Unicyclic Graphs With Minimal Distance Spectral Radius
The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) Cn.
Extreme distances in multicolored point sets.