Let G = (V,E) be a graph and f be a function f : V → {0, 1, 2}. A vertex u with f(u) = 0 is said to be undefended with respect to f , if it is not adjacent to a vertex with positive weight. The function f is a weak Roman dominating function (WRDF) if each vertex u with f(u) = 0 is adjacent to a vertex v with f(v) > 0 such that the function f ′ : V → {0, 1, 2} defined by f ′ (u) = 1, f ′ (v) = f...

We consider properties of connected diagrams with fermion-photon interaction and such fermion and photon propagators and vertex function that the values of these diagrams are finite. We establish the properties of these propagators and vertex function that imply that these diagrams are invariant under C, P, T, CP, CT, PT, or CPT transformations up to some phase factor common to each process. We...

recently, hua et al. defined a new topological index based on degrees and inverse ofdistances between all pairs of vertices. they named this new graph invariant as reciprocaldegree distance as 1{ , } ( ) ( ( ) ( ))[ ( , )]rdd(g) = u v v g d u  d v d u v , where the d(u,v) denotesthe distance between vertices u and v. in this paper, we compute this topological index forgrassmann graphs.

This paper proposes a game-theoretic approach to address the problem of optimal sensor placement for detecting cyber-attacks in networked control systems. The is formulated as zero-sum game with two players, namely malicious adversary and detector. Given protected target vertex, detector places at single vertex monitor system detect presence adversary. On other hand, selects through which condu...

An n-gon is defined as a sequence P = (V0, . . . , Vn−1) of n points on the plane. An n-gon P is said to be convex if the boundary of the convex hull of the set {V0, . . . , Vn−1} of the vertices of P coincides with the union of the edges [V0, V1], . . . , [Vn−1, V0]; if at that no three vertices of P are collinear then P is called strictly convex. We prove that an n-gon P with n > 3 is strictl...

A Roman domination function on a graph G is a function r : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman function is the value r(V (G)) = ∑ u∈V (G) r(u). The Roman domination number γR(G) of G is the minimum weight of a Roman domination function on G . "Roman Criticality" has been ...

A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection S of subsets of 1⁄2n 1⁄4 f1; 2...

the zagreb indices are the oldest graph invariants used in mathematical chemistry to predictthe chemical phenomena. in this paper we define the multiple versions of zagreb indicesbased on degrees of vertices in a given graph and then we compute the first and secondextremal graphs for them.

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

A vertex k-ranking of a graph G is a function c : V (G) → {1, . . . , k} such that if c(u) = c(v), u, v ∈ V (G) then each path connecting vertices u and v contains a vertex w with c(w) > c(u). If each vertex v has a list of integers L(v) and for a vertex ranking c it holds c(v) ∈ L(v) for each v ∈ V (G) then c is called L-list k-ranking, where L = {L(v) : v ∈ V (G)}. In this paper we investigat...