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

Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of $T$, then $G$ can be edge-decomposed into subgraphs isomorphic to $T$. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from $T$ by vertex-identifications. We call such a subgraph a homomorphic copy of $T$. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of $G$ is greater than the diameter of $T$. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page

Hypergraph polynomials and the Bernardi process

Recently O. Bernardi gave a formula for the Tutte polynomial $T(x,y)$ of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial $I$ is a generalization of $T(x,1)$ to hypergraphs. We supply a Bernardi-type description of $I$ using a ribbon structure on the underlying bipartite graph $G$. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of $G$ in the same way as $I$ is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses) constructed by Baker and Wang between spanning trees and break divisors.Comment: 46 page

No Laplacian Perfect State Transfer in Trees

We consider a system of qubits coupled via nearest-neighbour interaction governed by the Heisenberg Hamiltonian. We further suppose that all coupling constants are equal to $1$. We are interested in determining which graphs allow for a transfer of quantum state with fidelity equal to $1$. To answer this question, it is enough to consider the action of the Laplacian matrix of the graph in a vector space of suitable dimension. Our main result is that if the underlying graph is a tree with more than two vertices, then perfect state transfer does not happen. We also explore related questions, such as what happens in bipartite graphs and graphs with an odd number of spanning trees. Finally, we consider the model based on the $XY$-Hamiltonian, whose action is equivalent to the action of the adjacency matrix of the graph. In this case, we conjecture that perfect state transfer does not happen in trees with more than three vertices.Comment: 15 page