40 research outputs found
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree
AbstractIf G is a connected undirected simple graph on n vertices and n+c-1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c=3. Let Δ(G) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree Δ⩾n+c+12. Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when Δ(G)⩾1+6+2n32, and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices
A lower bound for the first Zagreb index and its application
For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index
The trace of uniform hypergraphs with application to Estrada index
In this paper we investigate the traces of the adjacency tensor of
hypergraphs (simply called the traces of hypergraphs). We give new expressions
for the traces of hypertrees and linear unicyclic hypergraphs by the weight
function assigned to their connected sub-hypergraphs, and provide some
perturbation results for the traces of a hypergraph with cut vertices. As
applications we determine the unique hypertree with maximum Estrada index among
all hypertrees with fixed number of edges and perfect matchings, and the unique
unicyclic hypergraph with maximum Estrada index among all unicyclic hypergraph
with fixed number of edges and girth