For an ordered set $W=\{w_1,w_2,...,w_k\}$ of $k$ distinct vertices in a nontrivial connected graph $G$, the metric code of a vertex $v$ of $G$ with respect to $W$ is the $k$-vector $$\mathrm{code}\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k))$$ where $d(v,w_i)$ is the distance between $v$ and $w_i$ for $1\le i\le k$. The set $W$ is a local metric set of $G$ if $\mathrm{code}\left(u\right)\ne \mathrm{code}\left(v\right)$ for every pair $u,v$ of adjacent vertices of $G$. The minimum positive integer $k$ for which $G$ has a local metric $k$-set is the local metric dimension $\mathrm{lmd}\left(G\right)$ of $G$. A local metric set of $G$ of cardinality $\mathrm{lmd}\left(G\right)$ is a local metric basis of $G$. We characterize all nontrivial connected...

In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipartite graphs play a significant role in designing very large interconnection networks. In this paper, we...

The path-distance-width of a connected graph G is the minimum integer w satisfying that there is a nonempty subset of S ⊆ V (G) such that the number of the vertices with distance i from S is at most w for any nonnegative integer i. In this note, we determine the path-distance-width of hypercubes.

The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like...

Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ Lesniak (1975) gave the lower bound $B(n,d)=\lceil 2(n-1)/d\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd, $B(n,d)$ is smaller than $L(n,d)$ in general. For example, $B(21,3)=14$ while $L(21,3)=19.$ In this note, we determine $L(n,d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

The Wiener index W(G) of a connected graph G, introduced by Wiener in 1947, is defined as W(G) = ∑u,v∈V(G) d(u, v) where dG(u, v) is the distance between vertices u and v of G. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and S ⊆ V (G), the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph whose vertex set is S. We now...

The strong isometric dimension of a reflexive graph is related to its injective hull: both deal with embedding reflexive graphs in the strong product of paths. We give several upper and lower bounds for the strong isometric dimension of general graphs; the exact strong isometric dimension for cycles and hypercubes; and the isometric dimension for trees is found to within a factor of two.

A method to infer $X$-trees (valued trees having $X$ as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2$n$-3 distance values between the $n$ elements of $X$, if they fulfill some explicit conditions. This construction is based on the mapping between $X$-tree and a weighted generalized 2-tree spanning $X$.

A method to infer X-trees (valued trees having X as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2n-3 distance values between the n elements of X, if they fulfill some explicit conditions. This construction is based on the mapping between X-tree and a weighted generalized 2-tree spanning X.

For a nontrivial connected graph $G$ of order $n$ and a linear ordering $s{v}_1,v_2,...,v_n$ of vertices of $G$, define $d\left(s\right)={\sum }_{i=1}^{n-1}d(v_i,v_{i+1})$. The traceable number $t\left(G\right)$ of a graph $G$ is $t\left(G\right)=min\left\{d\right(s\left)\right\}$ and the upper traceable number $t^{+}\left(G\right)$ of $G$ is $t^{+}\left(G\right)=max\left\{d\left(s\right)\right\},$ where the minimum and maximum are taken over all linear orderings $s$ of vertices of $G$. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs $G$ for which $t^{+}\left(G\right)-t\left(G\right)=1$ are characterized and a formula for the upper...

For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_D\left[S\right]={\bigcup }_{x,y\in S}{I}_D[x,y]$. A set S of vertices is a detour convex set if $I_D\left[S\right]=S$. The detour convex hull ${\left[S\right]}_D$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V(G) with ${\left[S\right]}_D=V\left(G\right)$....

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x -y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mₓ(G). An x-monophonic set of cardinality mₓ(G) is called a mₓ-set of G. We determine bounds for it and characterize graphs which realize these bounds. A connected graph of order p with vertex monophonic numbers...

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs having a distinguished or root vertex, labeled 0. The hierarchical product G2 ⊓ G1 of G2 and G1 is a graph with vertex set V2 × V1. Two vertices y2y1 and x2x1 are adjacent if and only if y1x1 ∈ E1 and y2 = x2; or y2x2 ∈ E2 and y1 = x1 = 0. In this paper, the Wiener, eccentric connectivity and Zagreb indices of this new operation of graphs are computed. As an application, these topological indices for a class of alkanes are computed. ACM Computing...

The Wiener number of a graph G is defined as 1/2∑d(u,v), where u,v ∈ V(G), and d is the distance function on G. The Wiener number has important applications in chemistry. We determine the Wiener number of an important family of graphs, namely, the Kneser graphs.

The Wiener number of a graph G is defined as $1/2{\sum }_{u,v\in V\left(G\right)}d(u,v)$, d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, $W\left(\mu \left(S{ₙ}^k\right)\right)\le W\left(\mu \left(T{ₙ}^k\right)\right)\le W\left(\mu \left(P{ₙ}^k\right)\right)$, where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of $\mu \left(G^k\right)$.