相关论文: Distance Magic Index One Graphs

A bijective function $f:V\rightarrow\left\{1,2,3,...,|V| \right\}$ is said to be a local distance antimagic labeling of a graph $G=(V,E)$, if $w(u)\neq w(v)$ for any two adjacent vertices $u, v$ where the weight $w(v)=\sum_{z\in N(v)}f(z)$.…

The support of a matrix M is the (0,1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M. This paper…

In this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of fixed order and given edge connectivity.

A connected graph is called a multi-block graph if each of its blocks is a complete multi-partite graph. Building on the work of \cite{Bp3,Hou3}, we compute the determinant and inverse of the distance matrix for a class of multi-block…

Distance-hereditary graphs are known to be the graphs that are totally decomposable for the split decomposition. We characterise distance-hereditary graphs whose complement is also distance-hereditary by their split decomposition and by…

A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs,…

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/…

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…

A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and…

The dodecacode is a nonlinear additive quaternary code of length $12$. By puncturing it at any of the twelve coordinates, we obtain a uniformly packed code of distance $5$. In particular, this latter code is completely regular but not…

A graph $G$ is called \textit{super edge-magic} if there exists a bijective function $f$ from $V(G) \cup E(G)$ to $\{1, 2, \ldots, |V(G) \cup E(G)|\}$ such that $f(V(G)) = \{1, 2, \ldots, |V(G)|\}$ and $f(x) + f(xy) + f(y)$ is a constant…

For a simple graph $G$, the $3$-distance graph, $D_3(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph $G$. For a connected graph $G$, we provide some conditions for…

We study the inertia of distance matrices of weighted graphs. Our novel congruence-based proof of the inertia of weighted trees extends to a proof for the inertia of weighted unicyclic graphs whose cycle is a triangle. Partial results are…

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

Let $G = (V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^d(G)=\sum_{v \in V_G}\,\varepsilon_G(v)D_G(v),$ where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v)=\sum_{u \in…

We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…

In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance…

The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix $D(G)$. A graph is called distance integral if all of its distance eigenvalues are integers. Let $n \geq 3$ be an integer. The crown graph $Cr(n)$…

Stanley's theorem establishes that for any finite graph $G$, the number $h_G(s)$ of magic labelings with magic sum $s$ can be expressed as a sum of two polynomials in $s$. However, determining the precise form of $h_G(s)$ is generally…