The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The edge grafting operation on a graph is a kind of edge moving between two vertices of the graph. In this paper, we introduce two new edge grafting operations and show how the graph energy changes under these edge grafting operations. Let G(n) be the set of all unicyclic graphs with n vertices. ...

Recent studies have shown that graph-based approaches are effective for semi-supervised learning. The key idea behind many graph-based approaches is to enforce the consistency between the class assignment of unlabeled examples and the pairwise similarity between examples. One major limitation with most graph-based approaches is that they are unable to explore dissimilarity or negative similarit...

The Detour matrix (DD) of a graph has for its ( i , j) entry the length of the longest path between vertices i and j. The DD-eigenvalues of a connected graph G are the eigenvalues for its detour matrix, and they form the DD-spectrum of G. The DD-energy EDD of the graph G is the sum of the absolute values of its DDeigenvalues. Two connected graphs are said to be DDequienergetic if they have equa...

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This “density conjecture” has been proved for the underlying graph Z. We ...

Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its c...

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. In the paper, we characterize the graphs with minimal energy in the class of bipartite unicyclic graphs of a given (p, q)–bipartition, where q ≥ p ≥ 2.

Given a complex m × n matrix A, we index its singular values as σ1 (A) ≥ σ2 (A) ≥ ... and call the value E (A) = σ1 (A)+σ2 (A)+ ... the energy of A, thereby extending the concept of graph energy, introduced by Gutman. Let 2 ≤ m ≤ n, A be an m×n nonnegative matrix with maximum entry α, and ‖A‖ 1 ≥ nα. Extending previous results of Koolen and Moulten for graphs, we prove that E (A) ≤ ‖A‖1 √ mn + √

For a simple graph G = (V , E) with eigenvalues of the adjacency matrix λ1 ≥ λ2 ≥ · · · ≥ λn, the energy of the graph is defined by E (G) = ∑n j=1 ⏐⏐λj⏐⏐. Myriads of papers have been published in the mathematical and chemistry literature about properties of this graph invariant due to its connection with the energy of (bipartite) conjugated molecules. However, a structural interpretation of thi...

Let G be a simple graph with vertex set V (G) = {v1, v2, . . . , vn} and edge set E(G) = {e1, e2, . . . , em}. Similar to the Randić matrix, here we introduce the Randić incidence matrix of a graph G, denoted by IR(G), which is defined as the n × m matrix whose (i, j)-entry is (di) 1 2 if vi is incident to ej and 0 otherwise. Naturally, the Randić incidence energy IRE of G is the sum of the sin...