More on energy and Randic energy of specific graphs

Let $G$ be a simple graph of order $n$. The energy $E(G)$ of the graph $G$ is the sum of the absolute values of the eigenvalues of $G$. The Randi\'{c} matrix of $G$, denoted by $R(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $(d_id_j)^{\frac{-1}{2}}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. The Randi\'{c} energy $RE$ of $G$ is the sum of absolute values of the eigenvalues of $R(G)$. In this paper we compute the energy and Randi\'{c} energy for certain graphs. Also we propose a conjecture on Randi\'c energy.Comment: 14 page

Algorithm and Complexity for a Network Assortativity Measure

We show that finding a graph realization with the minimum Randi\'c index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randi\'c index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randi\'c index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randi\'c index, our results extend to maximizing the Randi\'c index as well. Applications of the Randi\'c index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application

Randi\'c energy and Randi\'c eigenvalues

Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$ otherwise. The normalized signless Laplacian matrix $\mathcal{Q}$ is defined as $\mathcal{Q} =I+\bf{R}$, where $I$ is the identity matrix. The Randi\'c energy is the sum of absolute values of the eigenvalues of $\bf{R}$. In this paper, we find a relation between the normalized signless Laplacian eigenvalues of $G$ and the Randi\'c energy of its subdivided graph $S(G)$. We also give a necessary and sufficient condition for a graph to have exactly $k$ and distinct Randi\'c eigenvalues.Comment: 7 page

Upper Bounds for Randic Spread

The Randi´c spread of a simple undirected graph G, sprR(G), is equal to the maximal difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius [Gomes et al., MATCH Commun. Math. Comput. Chem. 72 (2014) 249–266]. Using a rank-one perturbation on the Randi´c matrix of G it is obtained a new matrix whose matricial spread coincide with sprR(G). By means of this result, upper bounds for sprR(G) are obtained