On the number of genus one labeled circle trees.
On the number of labeled -arch graphs.
On the Number of Minimal 1-Steiner Trees.
On the number of planar orientations with prescribed degrees.
On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition,...
On the rooted Tutte polynomial
The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph.
On the semigroup of fully indecomposable relations
On the theory of Pfaffian orientations. I: Perfect matchings and permanents.
On the theory of Pfaffian orientations. II: -joins, -cuts, and duality of enumeration.
On the unitary Cayley graph of a finite ring.
On upper bounds for total -domination number via the probabilistic method
For a fixed positive integer and a connected graph of order , whose minimum vertex degree is at least , a set is a total -dominating set, also known as a -tuple total dominating set, if every vertex has at least neighbors in . The minimum size of a total -dominating set for is called the total -domination number of , denoted by . The total -domination problem is to determine a minimum total -dominating set of . Since the exact problem is in general quite difficult to solve,...