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...

Eccentric distance sum (EDS), which can predict biological and physical properties, is a topological index based on the eccentricity of a graph. In this paper we characterize the chain hexagonal cactus with the minimal and the maximal eccentric distance sum among all chain hexagonal cacti of length n, respectively. Moreover, we present exact formulas for EDS of two types of hexagonal cacti.

The effect of retinal eccentricity on stimulus identification was investigated by measuring observers' ability to identify digits of various sizes presented briefly at different locations in their visual fields. The stimuli were either presented one at a time or presented in pairs with each member of the pair at a different eccentricity and having a different size. Stimulus-response confusion m...

Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called its spectrum. There are many properties which can be explained using the spectrum like energy, connectedness, vertex connectivity, chromatic number, perfect matching etc. Laplacian spectrum is the multiset of eigenvalues of Laplacian matrix. The Laplacian energy of a graph is the sum of the...

When a square matrix A, is diagonalizable, (for example, when A is Hermitian or has distinct eigenvalues), then An can be written as a sum of the nth powers of its eigenvalues with matrix weights. However, if a 1 occurs in its Jordan form, then the form is more complicated: An can be written as a sum of polynomials of degree n in its eigenvalues with coefficients depending on n. In this case to...

In a connected graph G, the distance between two vertices of G is length shortest path these vertices. The eccentricity vertex u in largest and any other G. total-eccentricity index ?(G) sum eccentricities all this paper, we find extremal trees, unicyclic bicyclic graphs with respect to index. Moreover, conjugated trees

The energy of signed graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two signed graphs are said to be equienergetic if they have same energy. In the literature the construction of equienergetic signed graphs are reported. In this paper we obtain the characteristic polynomial and energy of the join of two signed graphs and thereby we give another construction ...