The concept of average degree-eccentricity matrix ADE(G) of a connected graph $G$ is introduced. Some coefficients of the characteristic polynomial of ADE(G) are obtained, as well as a bound for the eigenvalues of ADE(G). We also introduce the average degree-eccentricity graph energy and establish bounds for it.

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and the average degree of the vertices of G. Motivated by the work from Sharafdini an...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. 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 ext...

The energy of a graph G is equal to the sum of absolute values of the eigenvalues of the adjacency matrix of G, whereas the Laplacian energy of a graph G is equal to the sum of the absolute value of the difference between the eigenvalues of the Laplacian matrix of G and average degree of the vertices of G. Motivated by the work from Sharafdini et al. [R. Sharafdini, H. Panahbar, Vertex weighted...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. In this paper we resolve five conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (independence number, chromatic number and the Randić index), and refute t...