The local adjacency polynomials can be thought of as a generalization, for all graphs, of (the sums of ) the distance polynomials of distance-regular graphs. The term ``local'' here means that we ``see'' the graph from a given vertex, and it is the price we must pay for speaking of a kind of distance-regularity when the graph is not regular. It is shown that when the value at * (the maximum eig...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

The concept of local pseudo-distance-regularity, introduced in this paper, can be thought of as a natural generalization of distance-regularity for non-regular graphs. Intuitively speaking, such a concept is related to the regularity of graph 1 when it is seen from a given vertex. The price to be paid for speaking about a kind of distance-regularity in the non-regular case seems to be locality....

Sierpiński graphs S p form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of S p is introduced as a vertex that is either adjacent to an extreme vertex of S p or is incident to an edge between two subgraphs of S p isomorphic to S n−1 p . Explicit formulas are given for...

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph Γ, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular...

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, . . . , fd} of k-distance Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 2 for each v ∈ V (G), is call...

Let G be a digraph. For two vertices u and v in G, the distance d(u, v) from u to v in G is the length of the shortest directed path from u to v. The eccentricity e(v) of v is the maximum distance of v to any other vertex of G. A vertex u is an eccentric vertex of v if the distance from v to u is equal to the eccentricity of v. The eccentric digraph ED(G) of G is the digraph that has the same v...

A set $S = {u_1,u_2, ldots, u_t}$ of vertices of $G$ is an efficientdominating set if every vertex of $G$ is dominated exactly once by thevertices of $S$. Letting $U_i$ denote the set of vertices dominated by $u_i$%, we note that ${U_1, U_2, ldots U_t}$ is a partition of the vertex setof $G$ and that each $U_i$ contains the vertex $u_i$ and all the vertices atdistance~1 from it in $G$. In this ...

Let η(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper, bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimize...

Let G be an infinite geometric graph; in particular, a graph whose vertices are a countable discrete set of points on the plane, with vertices u, v adjacent if their Euclidean distance is less than 1. A “fire” begins at some finite set of vertices and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time-step. Let f(G) be the least f for which any fi...