Narrow Results

Let $G$ be a non-abelian group and $V\subseteq G.$ We define the commuting graph $\mathrm{\Gamma }=\zeta (G,V)$ with vertex set $V$ and two distinct vertices $u$ and $v$ of $V$ are connected by an edge when they commute in $G$. In this article, certain distance and detour distance properties of commuting graph of non-abelian groups of order $p^4$ with center having $p$ elements, where $p$ is an odd prime have been studied. Metric dimensions, resolving polynomials and spectral properties of such graphs have also been effectively computed.

Let $G$ be a $(p$, $q)$- graph, $D(G)$ the detour diameter of $G$, and cdn$(G)$ is the connected detour number of $G$. Santhakumaran and Athisayanathan are characterized by the graphs of $D(G)\le$ 4 and cdn$(G)=p$, $p-1$ and $p-2$. In this paper, we obtain a necessary and sufficient condition for a graph $G$, $p(G)\ge$ 2, of $D(G)\le$ 8 to have cdn$(G)=p$. Moreover, graphs G of $p(G)\ge$ 6, and $D(G)$ = 5 and 6, such that cdn$(G)=p-1$, are characterized in this research.

The length of a longest $u-v$ path in a connected graph $G$ is called detour distance and denoted by $D(u,v)$, where the vertices $u$ and $v$ belong to the vertex set $V(G)$. In this paper, we find detour polynomial and detour index of $n$ connected vertex disjoint graphs, for $n\ge 3$ by the vertices identified and edges introducing with isolated vertex. Also, we find detour polynomial and detour index for some compound graphs from special graphs.

In this paper, we define the clique-to-vertex $C$–$v$ monophonic path, the clique-to-vertex monophonic distance $d_m(C,v)$, the clique-to-vertex monophonic eccentricity $e_{m_2}(C)$, the clique-to-vertex monophonic radius $R_{m_2}$, and the clique-to-vertex monophonic diameter $D_{m_2}$, where $C$ is a clique and $v$ a vertex in a connected graph $G$. We determine these parameters for some standard graphs. We show the inequality among the clique-to-vertex distance, the clique-to-vertex monophonic distance, and the clique-to-vertex detour distance in graphs. Also, it is shown that the clique-to-vertex geodesic, the clique-to-vertex monophonic, and the clique-to-vertex detour are distinct in $G$. It is shown that $R_{m_2}\le D_{m_2}$ for every connected graph $G$ and that every two positive integers $a$ and $b$ with $2\le a\le b$ are realizable as the clique-to-vertex monophonic radius and clique-to-vertex monophonic diameter of some connected graph. Also, it is shown any three positive integers $a,b,c$ with $3\le a\le b\le c$ are realizable as the clique-to-vertex radius, clique-to-vertex monophonic radius, and clique-to-vertex detour radius of some connected graph and also it is shown that any three positive integers $a,b,c$ with $4\le a\le b\le c$ are realizable as the clique-to-vertex diameter, clique-to-vertex monophonic diameter, and clique-to-vertex detour diameter of some connected graph. We introduce the clique-to-vertex monophonic center $C_{m_2}(G)$ and the clique-to-vertex monophonic periphery $P_{m_2}(G)$ and it is shown that the clique-to-vertex monophonic center does not lie in a single block of $G$.

A set $S\subseteq V$ is called an open detour set of $G$ if for each vertex $v$ in $G$, either (1) $v$ is a detour simplicial vertex of $G$ and $v\in S$ or (2) $v$ is an internal vertex of an $x$-$y$ detour for some $x,y\in S$. An open detour set of minimum cardinality is called a minimum open detour set and this cardinality is the open detour number of $G$, denoted by $odn(G)$. Connected graphs of order $p$ with open detour number $p$ or $p-1$ are characterized. It is shown that for any two positive integers $a$ and $b$ with $b\ge 2a$, there exists a connected graph $G$ such that $dn(G)=b$ and $odn(G)=a+b$, where $dn(G)$ is the detour number of $G$. It is also shown that for every pair of positive integers $a$ and $b$ with $a\ge 4$ and $b\ge 4$, there exists a connected graph $G$ such that $og(G)=a$ and $odn(G)=b$, where $og(G)$ is the open geodetic number of $G$.