Related papers: On Eccentricity Matrices of Wheel Graphs

Let $G$ be a connected graph on $n$ vertices and $D(G)$ its distance matrix. The formula for computing the determinant of this matrix in terms of the number of vertices is known when the graph is either a tree or {a} unicyclic graph. In…

Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the…

Let $G=(V_G,E_G)$ be a connected graph. The distance $d_G(u,v)$ between vertices $u$ and $v$ in $G$ is the length of a shortest $u-v$ path in $G$. The eccentricity of a vertex $v$ in $G$ is the integer $e_G(v)= \max\{ d_G(v,u) \colon u\in…

A set $W\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum…

For a nonabelian group G, the non-commuting graph $\Gamma_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_G$ are adjacent if they do not commute in $G$. In…

The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equation $W(G)=W(G-v)$…

In this paper, we aim at recovering an undirected weighted graph of $N$ vertices from the knowledge of a perturbed version of the eigenspaces of its adjacency matrix $W$. For instance, this situation arises for stationary signals on graphs…

Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…

The Helmholtzian matrix of a graph $G=(V(G),E(G))$ is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) [S. Li, L. Lu, J.F. Wang, A graph discretization of vector Laplacian, 379 (2026) 446--460]. Motivated by the…

The eccentric-connectivity index of a graph G is the sum of the products of the eccentricity and the degree of each vertex in G. In this paper, we define four new invariants related to the eccentric-connectivity index and obtain upper…

We study the inverse eigenvector centrality problem on connected undirected graphs, namely, whether a given positive vector can be realized by assigning suitable edge weights. We provide a complete characterization in terms of stable sets…

The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the…

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound.…

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of…

In this paper, we introduce the Laplacian and the signless Laplacian for the eccentricity matrix of a connected graph, referred to as the eccentricity Laplacian and the eccentricity signless Laplacian, respectively. We establish the…

Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_R$ of $R$ is a graph in which the vertex set comprises of set of all nonzero proper ideals of $R$ and two vertices $I$ and $K$ are adjacent if and only if…

Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum.…

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral…