This paper determines all nonsingular unicyclic mixed graphs on at least nine vertices with at most three Laplacian eigenvalues greater than two.

The Wiener index of a graph G is defined as W (G) = ∑ u,v dG(u, v), where dG(u, v) is the distance between u and v in G, and the sum goes over all pairs of vertices. In this paper, we characterize the connected unicyclic graph with minimum Wiener indices among all connected unicyclic graphs of order n and girth g with k pendent vertices.

We describe an O(n log n) algorithm for the computation of the vertex separation of unicyclic graphs. The algorithm also computes a linear layout with optimal vertex separation in the same time bound. Pathwidth, node search number and vertex separation are different ways of defining the same notion. Path decompositions and search strategies can be derived from linear layouts. The algorithm appl...

This paper surveys extremal properties of general sum-connectivity index χα(G) in several classes of connected graphs of given order for some values of the parameter α: a) trees; b) connected unicyclic or bicyclic graphs; c) graphs of given connectivity. Work presented as invited lecture at CAIM 2014, September 19-22, “Vasile Alecsandri” University of Bacău, Romania.

We introduce a new operation on a class of graphs with the property that the Laplacian eigenvalues of the input and output graphs are related. Based on this operation, we obtain a family of Θ( √ n) noncospectral unicyclic graphs on n vertices with the same Laplacian energy.