the reverse degree distance of a connected graph $g$ is defined in discrete mathematical chemistry as [ r (g)=2(n-1)md-sum_{uin v(g)}d_g(u)d_g(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $g$, respectively, $d_g(u)$ is the degree of vertex $u$, $d_g(u)$ is the sum of distance between vertex $u$ and all other vertices of $g$, and $v(g)$ is the ...

the total irregularity of a graph g is defined as 〖irr〗_t (g)=1/2 ∑_(u,v∈v(g))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈v(g). in this paper by using the gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

‎the harmonic index of a connected graph $g$‎, ‎denoted by $h(g)$‎, ‎is‎ ‎defined as $h(g)=sum_{uvin e(g)}frac{2}{d_u+d_v}$‎ ‎where $d_v$ is the degree of a vertex $v$ in g‎. ‎in this paper‎, ‎expressions for the harary indices of the‎ ‎join‎, ‎corona product‎, ‎cartesian product‎, ‎composition and symmetric difference of graphs are‎ ‎derived‎.

Let $G$ be a simple graph with vertex set ${v_1,v_2,ldots,v_n}$. The common neighborhood graph (congraph) of $G$, denoted by $con(G)$, is the graph with vertex set ${v_1,v_2,ldots,v_n}$, in which two vertices are adjacent if and only they have at least one common neighbor in the graph $G$. The basic properties of $con(G)$ and of its energy are established.

For a median graph G and a vertex v of G that is not a cut-vertex we show that G − v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whos...

Proof. “⇐” Suppose G has a cut vertex v. Then Gr v has at least two components G1 and G2. Given vertices vi ∈ V (Gi), i = 1, 2. Each path connecting v1 and v2 must contain the vertex v. So any two paths between v1 and v2 must contain the common internal vertex v. This is a contradiction. “⇒” Given two distinct vertices u and v of G, let d(u, v) denote the distance between u and v. We apply indu...

Let G be a connected graph and g non-negative integer. A vertex set S of is called g-extra cut if G−S disconnected each component has at least g+1 vertices. The connectivity the minimum cardinality one cut. For two graphs G1=(V1,E1) G2=(V2,E2), strong product G1⊠G2 defined as follows: its V1×V2 edge {(x1,x2)(y1,y2)|x1=x2 y1y2∈E2; or y1=y2 x1x2∈E1; x1x2∈E1 y1y2∈E2}, where (x1,x2),(y1,y2)∈V1×V2. ...

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperti...

The total irregularity of a graph G is defined as 〖irr〗_t (G)=1/2 ∑_(u,v∈V(G))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈V(G). In this paper by using the Gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.