This thesis provides an alternate proof of the Matrix Tree Theorem by shifting the focus to oriented incidences. We examine the weak walk contributors from the determinant of the Laplacian matrix of oriented graphs and classify them according to similar circle structures attained through circle activation. The members of each of these contribution classes form an alternating rank-signed Boolean...

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...

A concept related to the spectrum of a graph is that of energy. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G . The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues of G and the average degree d(G) of G. In this paper we introduce the concept of Laplacian energy of fuzzy graphs. ...

In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectral evolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular value decomposition. The spectral evolution model states that over time, the eigenvalues of a network change while its eig...

We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions ...