A Characterization of the Degree Sequences of 2-Trees

A graph G is a 2-tree if G=K_3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v{i}s a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees.Comment: 17 pages, 5 figure

Spectral Bounds for the Connectivity of Regular Graphs with Given Order

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, we present two upper bounds for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.Comment: 24 page

Degree Sequences of Edge-Colored Graphs in Specified Families and Related Problems

Movement has been made in recent times to generalize the study of degree sequences to k-edge-colored graphs and doing so requires the notion of a degree vector\u3c\italic\u3e. The degree vector of a vertex v\u3c\italic\u3e in a k-edge-colored graph is a column vector in which entry i\u3c\italic\u3e indicates the number of edges of color i\u3c\italic\u3e incident to v\u3c\italic\u3e. Consider the following question which we refer to as the \u3c\italic\u3ek-Edge-Coloring Problem\u3c\italic\u3e. Given a set of column vectors C\u3c\italic\u3e and a graph family F\u3c\italic\u3e, when does there exist some k-edge-colored graph in F\u3c\italic\u3e whose set of degree vectors is C\u3c\italic\u3e? This question is NP-Complete in general but certain graph families yield tractable results. In this document, I present results on the k-Edge-Coloring Problem and the related Factor Problem for the following families of interest: unicyclic graphs, disjoint unions of paths (DUPs), disjoint union of cycles (DUCs), grids, and 2-trees

Fibonacci graphs

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant omega to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for 1 <= n <= 4

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Bond Additive Modeling 1. Adriatic Indices

Some of the most famous molecular descriptors are bond additive, i.e. they are calculated as the sum of edge contributions (Randić-type indices, Balaban-type indices, Wiener index and its modifications, Szeged index...). In this paper, the methods of calculations of bond contributions of these descriptors are analyzed. The general concepts are extracted, and based on these concepts a large class of molecular descriptors is defined. These descriptors are named Adriatic indices. An especially interesting subclass of these descriptors consists of 148 discrete Adriatic indices. They are analyzed on the testing sets provided by the International Academy of Mathematical Chemistry, and it has been shown that they have good predictive properties in many cases. They can be easily encoded in the computer and it may be of interest to incorporate them in the existing software packages for chemical modeling. It is possible that they could improve various QSAR and QSPR studies