Related papers: On adjacency and Laplacian cospectral non-isomorph…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
It is shown that an undirected graph $G$ is cospectral with the Hermitian adjacency matrix of a mixed graph $D$ obtained from a subgraph $H$ of $G$ by orienting some of its edges if and only if $H=G$ and $D$ is obtained from $G$ by a…
Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. Assuming rational independence of edge lengths, necessary and sufficient…
In this partly expository paper we discuss and describe some of our old and recent results on partial orders on the set (m,n)-graphs (i.e. graphs with n vertices and m edges) and some operations on graphs that are monotone with respect to…
The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this…
In this paper, we investigate spectral properties of the adjacency tensor, Laplacian tensor and signless Laplacian tensor of general hypergraphs (including uniform and non-uniform hypergraphs). We obtain some bounds for the spectral radius…
We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs.…
We study two different types of gluing for graphs: interface (obtained by choosing a common subgraph as the gluing component) and bridge gluing (obtained by adding a set of edges to the given subgraphs). We introduce formulae for computing…
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or…
In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also…
We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the…
Fix $m \in \mathbb N$. A new generalization of the $H$-join operation of a family of graphs $\{G_1, G_2, \dots, G_k\}$ constrained by indexing maps $I_1,I_2,\dots,I_k$ is introduced as $H_m$-join of graphs, where the maps $I_i:V(G_i)$ to…
Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The subdivision graph $\mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex…
Let $S$ be a set of transpositions generating the symmetric group $S_n$. The transposition graph of $S$ is defined to be the graph with vertex set $\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$ whenever $(i,j) \in…
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of…
A signed graph $\Sigma = (G, \sigma)$ is a graph where the function $\sigma$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $\Sigma$, denoted by $A(\Sigma)$, is defined canonically. In a recent…
A graph is determined by its signless Laplacian spectrum if there is no other non-isomorphic graph sharing the same signless Laplacian spectrum. Let $C_l$, $P_l$, $K_l$ and $K_{s,l-s}$ be the cycle, the path, the complete graph and the…
For a graph $G$ and a non-zero real number $\alpha$, the graph invariant $S_{\alpha}(G)$ is the sum of the $\alpha^{th}$ power of the non-zero signless Laplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of…
We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights…
There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree…