On Energy, Laplacian Energy and PP-fold Graphs

For a graph $G$ having adjacency spectrum ($A$-spectrum) $\lambda_n\leq\lambda_{n-1}\leq\cdots\leq\lambda_1$ and Laplacian spectrum ($L$-spectrum) $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$, the energy is defined as $E(G)=\sum_{i=1}^{n}|\lambda_i|$ and the Laplacian energy is defined as $LE(G)=\sum_{i=1}^{n}|\mu_i-\frac{2m}{n}|$. In this paper, we give upper and lower bounds for the energy of $KK_n^j,~1\leq j \leq n$ and as a consequence we generalize a result of Stevanovic et al. [More on the relation between Energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. {\bf 61} (2009) 395-401]. We also consider strong double graph and strong $p$-fold graph to construct some new families of graphs $G$ for which E(G)> LE(G)

Energy, Laplacian energy of double graphs and new families of equienergetic graphs

For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \cdots, v_n\}$, the extended double cover $G^*$ is a bipartite graph with bipartition (X, Y), $X=\{x_1, x_2, \cdots, x_n\}$ and $Y=\{y_1, y_2, \cdots, y_n\}$, where two vertices $x_i$ and $y_j$ are adjacent if and only if $i=j$ or $v_i$ adjacent to $v_j$ in $G$. The double graph $D[G]$ of $G$ is a graph obtained by taking two copies of $G$ and joining each vertex in one copy with the neighbours of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs $G^*$ and $D[G]$, $L$-spectra of $G^{k*}$ the $k$-th iterated extended double cover of $G$. We obtain a formula for the number of spanning trees of $G^*$. We also obtain some new families of equienergetic and $L$-equienergetic graphs.Comment: 23 pages, 1 figur

Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem

The conventional theory of solids is well suited to describing band structures locally near isolated points in momentum space, but struggles to capture the full, global picture necessary for understanding topological phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298 (2017)], we have introduced the way to overcome this difficulty by formulating the problem of sewing together many disconnected local "k-dot-p" band structures across the Brillouin zone in terms of graph theory. In the current manuscript we give the details of our full theoretical construction. We show that crystal symmetries strongly constrain the allowed connectivities of energy bands, and we employ graph-theoretic techniques such as graph connectivity to enumerate all the solutions to these constraints. The tools of graph theory allow us to identify disconnected groups of bands in these solutions, and so identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529 v2: Accepted version, minor typos corrected and references added. Now 19+epsilon page

Graphs and networks theory

This chapter discusses graphs and networks theory

Spectral Characterization of functional MRI data on voxel-resolution cortical graphs

The human cortical layer exhibits a convoluted morphology that is unique to each individual. Conventional volumetric fMRI processing schemes take for granted the rich information provided by the underlying anatomy. We present a method to study fMRI data on subject-specific cerebral hemisphere cortex (CHC) graphs, which encode the cortical morphology at the resolution of voxels in 3-D. We study graph spectral energy metrics associated to fMRI data of 100 subjects from the Human Connectome Project database, across seven tasks. Experimental results signify the strength of CHC graphs' Laplacian eigenvector bases in capturing subtle spatial patterns specific to different functional loads as well as experimental conditions within each task.Comment: Fixed two typos in the equations; (1) definition of L in section 2.1, paragraph 1. (2) signal de-meaning and normalization in section 2.4, paragraph