The main result: If the vertices of a connected graph are labelled by positive real numbers such that the number assigned to any vertex is half of the sum of the numbers assigned to the vertices of its neighbourhood, then each label is an integral multiple of the minimum of all labels. Using this, a result proved earlier in [7] is derived: If \(V\) is a linearly dependent subset of a root system in which all roots have the same norm, then one of the roots in \(V\) is an integral combination of the other roots in \(V\).

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is said to be a dominating set of \(G\) if each \(v \in V – D\) is adjacent to at least one vertex of \(D\). The minimum cardinality of a dominating set of \(G\) is called the domination number of \(G\) and is denoted by \(\gamma(G)\). A dominating set \(D\) with cardinality \(\gamma(G)\) is called a \(\gamma\)-set of \(G\). Given a graph \(G\), a new graph, denoted by \(\gamma.G\) and called the \(\gamma\)-graph of \(G\), is defined as follows: \(V(\gamma.G)\) is the set of all \(\gamma\)-sets of \(G\) and two sets \(D\) and \(S\) of \(V(\gamma.G)\) are adjacent in \(\gamma.G\) if and only if \(|D \cap S| = \gamma(G) – 1\). A graph \(G\) is said to be \(\gamma\)-connected if \(\gamma.G\) is connected. A graph \(G\) is said to be a \(\gamma\)-graph if there exists a graph \(H\) such that \(\gamma-H\) is isomorphic to \(G\). In this paper, we show that trees and unicyclic graphs are \(\gamma\)-graphs. Also, we obtain a family of graphs which are not \(\gamma\)-graphs.

A Cayley graph is a graph constructed out of a group \(\Gamma\) and its generating set \(A\). In this paper, we determine the independent domination number, perfect domination number, and independent dominating sets of \(Cay(\mathbb{Z}_n, A)\), for a specified generating set \(A\) of \(\mathbb{Z}_n\).

In this paper, we introduce an online tessellation partial automaton to recognize partial array languages. We also introduce two classes of partial array languages. We also introduce two classes of partial array languages viz, Local Partial Array Languages (PAL-LOC) Recognizable Partial Array Languages (PAL-REC) and prove PAL-REC is exactly the family of partial array languages recognizable by online tessellation partial automaton.

For a connected graph \(G\) of order \(p \geq 2\), a set \(S \subseteq V(G)\) is a geodetic set of \(G\) if each vertex \(v \in V(G)\) lies on an \(x\)-\(y\) geodesic for some elements \(x\) and \(y\) in \(S\). The minimum cardinality of a geodetic set of \(G\) is defined as the geodetic number of \(G\), denoted by \(g(G)\). A geodetic set of cardinality \(g(G)\) is called a \(g\)-set of \(G\). A connected geodetic set of \(G\) is a geodetic set \(S\) such that the subgraph \(G[S]\) induced by \(S\) is connected. The minimum cardinality of a connected geodetic set of \(G\) is the connected geodetic number of \(G\) and is denoted by \(g_c(G)\). A connected geodetic set of cardinality \(g_c(G)\) is called a \(g_c\)-set of \(G\). Connected graphs of order \(p\) with connected geodetic number \(2\) or \(p\) are characterized. It is shown that for positive integers \(r,d\) and \(n \geq d+1\) with \(r \leq d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(g_c(G) = n\). Also, for integers \(p,d\) and \(n\) with \(2 \leq d \leq p-1\), \(d+1 \leq n \leq p\), there exists a connected graph \(G\) of order \(p\), diameter \(d\) and \(g_c(G) = n\).

For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the \({detour\; distance}\) \(D(u, v)\) is the length of a longest \(u\)-\(v\) path in \(G\). A \(u\)-\(v\) path of length \(D(u,v)\) is called a \(u\)-\(v\) detour. A set \(S \subseteq V\) is called a \({detour \;set}\) of \(G\) if every vertex in \(G\) lies on a detour joining a pair of vertices of \(S\). The \({detour \;number}\) \(dn(G)\) of \(G\) is the minimum order of its detour sets, and any detour set of order \(dn(G)\) is a detour basis of \(G\). A set \(S \subseteq V\) is called a \({connected \;detour \;set}\) of \(G\) if \(S\) is a detour set of \(G\) and the subgraph \(G[S]\) induced by \(S\) is connected. The \({connected\; detour\; number}\) \(cdn(G)\) of \(G\) is the minimum order of its connected detour sets, and any connected detour set of order \(cdn(G)\) is called a \({connected\; detour \;basis}\) of \(G\). Certain general properties of these concepts are studied. The connected detour numbers of certain classes of graphs are determined. The relationship of the connected detour number with the detour diameter is discussed, and it is proved that for each triple \(D, k, p\) of integers with \(3 \leq k \leq p-D-1\) and \(D \geq 4\), there is a connected graph \(G\) of order \(p\) with detour diameter \(D\) and \(cdn(G) = k\). A connected detour set \(S\), no proper subset of which is a connected detour set, is a \({minimal\; connected\; detour\; set}\). The \({upper\; connected \;detour\; number}\) \(cdn^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal connected detour set of \(G\). It is shown that for every pair \(a, b\) of integers with \(5 \leq a \leq b\), there is a connected graph \(G\) with \(cdn(G) = a\) and \(cdn^+(G) = b\).

For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the \({detour\; distance}\) \(D(u,v)\) is the length of a longest \(u\)-\(v\) path in \(G\). A \(u\)-\(v\) path of length \(D(u, v)\) is called a \(u\)-\(v\) \({detour}\). A set \(S \subseteq V\) is called an \({edge\; detour \;set}\) if every edge in \(G\) lies on a detour joining a pair of vertices of \(S\). The \({edge \;detour\; number}\) \(dn_1(G)\) of \(G\) is the minimum order of its edge detour sets, and any edge detour set of order \(dn_1(G)\) is an \({edge\; detour\; basis}\) of \(G\). A connected graph \(G\) is called an \({edge\; detour\; graph}\) if it has an edge detour set. Certain general properties of these concepts are studied. The edge detour numbers of certain classes of graphs are determined. We show that for each pair of integers \(k\) and \(p\) with \(2 \leq k < p\), there is an edge detour graph \(G\) of order \(p\) with \(dn_1(G) = k\). An edge detour set \(S\), no proper subset of which is an edge detour set, is a \({minimal\; edge \;detour\; set}\). The \({upper\; edge\; detour\; number}\) \(dn_1^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal edge detour set of \(G\). We determine the upper edge detour numbers of certain classes of graphs. We also show that for every pair \(a, b\) of integers with \(2 \leq a \leq 6\), there is an edge detour graph \(G\) with \(dn_1(G) = a\) and \(dn_1^+(G) = b\).

An orthogonal double cover (ODC) of the complete graph \(K_n\) is a collection \(\mathcal{G} = \{G_1,G_2,\ldots,G_n\}\) of \(n\) subgraphs of \(K_n\), such that every edge of \(K_n\) belongs to exactly two of the \(G_i\)’s and every pair of \(G_i\)’s intersect in exactly one edge. If \(G_i \cong G\) for all \(i \in \{1,2,\ldots,n\}\), then \(\mathcal{G}\) is an ODC of \(K_n\) by \(G\). An ODC of \(K_n\) is \({cyclic}\) (CODC) if the cyclic group of order \(n\) is a subgroup of its automorphism group. In this paper, we find CODCs of complete graphs by the complete multipartite graphs \(K_{2,r,s}\), \(K_{1,1,r,s}\), and \(K_{1,1,1,1,r}\).

An \({Edge\; Roman\; dominating\; function}\) of a graph \(G = (V, E)\) is a function \(f’ : E \to \{0,1,2\}\) satisfying the condition that every edge \(x\) for which \(f'(x) = 0\) is adjacent to at least one edge \(y\) for which \(f'(y) = 2\). The \({weight}\) of an Edge Roman dominating function is the value \(f'(E) = \sum_{x\in E} f'(x)\). The minimum weight of an Edge Roman dominating function on a graph \(G\) is called the \({Edge\; Roman\; domination\; number}\) of \(G\). In this paper, we initiate a study of this parameter.

The energy \(E(G)\) of a graph \(G\) is the sum of the absolute values of the eigenvalues of \(G\). Two graphs \(G_1\) and \(G_2\) are said to be equienergetic if \(E(G_1) = E(G_2)\). In this paper, we outline various classes of equienergetic graphs. These results enable the construction of pairs of noncospectral equienergetic graphs of the same order and of the same size.