We compare the asymptotic growth of the order of the digraphs arising from a construction of Comellas and Fiol when applied to Faber-Moore digraphs versus plainly the Faber-Moore digraphs for the corresponding degree and diameter.

We derive the asymptotic spectral distribution of the distance k-graph of N-dimensional hypercube as N → ∞.

Let G be a finite connected graph on two or more vertices, and $G^{[N,k]}$ the distance-k graph of the N-fold Cartesian power of G. For a fixed k ≥ 1, we obtain explicitly the large N limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of $G^{[N,k]}$. The limit distribution is described in terms of the Hermite polynomials. The proof is based on asymptotic combinatorics along with quantum probability theory.

In this paper, we develop a divide-and-conquer approach, called block decomposition, to solve the minimum geodetic set problem. This provides us with a unified approach for all graphs admitting blocks for which the problem of finding a minimum geodetic set containing a given set of vertices (g-extension problem) can be efficiently solved. Our method allows us to derive linear time algorithms for the minimum geodetic set problem in (a proper superclass of) block-cacti and monopolar chordal graphs....

Formulas for vertex eccentricity and radius for the n-fold tensor product $G={\otimes }_{i=1}ⁿG_i$ of n arbitrary simple graphs $G_i$ are derived. The center of G is characterized as the union of n+1 vertex sets of form V₁×V₂×...×Vₙ, with $V_i\subseteq V\left(G_i\right)$.

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r\left(v\right|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the metric representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between vertices $x$ and $y$. A set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we characterize all graphs of order $n$ with metric dimension $n-3$.

In a graph G, the distance d(u, v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius, r(G), of the graph and the maximum eccentricity is called the diameter, d(G), of the graph. The super-radial graph R*(G) based on G has the vertex set as in G and two vertices u and v are adjacent in R*(G) if the distance between them in G is greater than...

As was shown in the book of Mulder [4], the interval function is an important tool for studying metric properties of connected graphs. An axiomatic characterization of the interval function of a connected graph was given by the present author in [5]. (Using the terminology of Bandelt, van de Vel and Verheul [1] and Bandelt and Chepoi [2], we may say that [5] gave a necessary and sufficient condition for a finite geometric interval space to be graphic). In the present paper, the result given in [5]...

In this paper, we show that Qkn is a divisor graph, for n = 2, 3. For n ≥ 4, we show that Qkn is a divisor graph iff k ≥ n − 1. For folded-hypercube, we get FQn is a divisor graph when n is odd. But, if n ≥ 4 is even integer, then FQn is not a divisor graph. For n ≥ 5, we show that (FQn)k is not a divisor graph, where 2 ≤ k ≤ [n/2] − 1.

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Circular distance $d^{\circ }(x,y)$ between two vertices $x$, $y$ of a strongly connected directed graph $G$ is the sum $d(x,y)+d(y,x)$, where $d$ is the usual distance in digraphs. Its basic properties are studied.

The eccentricity of a vertex $v$ of a connected graph $G$ is the distance from $v$ to a vertex farthest from $v$ in $G$. The center of $G$ is the subgraph of $G$ induced by the vertices having minimum eccentricity. For a vertex $v$ in a 2-edge-connected graph $G$, the edge-deleted eccentricity of $v$ is defined to be the maximum eccentricity of $v$ in $G-e$ over all edges $e$ of $G$. The edge-deleted center of $G$ is the subgraph induced by those vertices of $G$ having minimum edge-deleted eccentricity. The edge-deleted central...