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.

In this paper, fuzzy finite state automaton with unique membership transition on an input symbol (uffsa) is defined. It is proved and illustrated that for a given fuzzy finite state automaton (ffsa), there exists an equivalent uffsa. Some closure properties of fuzzy regular languages are studied.

A \((G,H)\)-multifactorization of \(\lambda K_m\) is a partition of the edge set of \(\lambda K_m\) into \(G\)-factors and \(H\)-factors with at least one \(G\)-factor and one \(H\)-factor. Atif Abueida and Theresa O’Neil have conjectured that for any integer \(n \geq 3\) and \(m \geq n\), there is a \((G_n, H_n)\)-multidecomposition of \(\lambda K_m\) where \(G_n = K_{1,n-1}\) and \(H_n = C_n\). In this paper, it is shown that the above conjecture is true for \(m=n\) when

1. \(G_m = K_{1,m-1}; H_m = C_m\),
2. \(G_m = H_{1,m-1}; H_m = P_m\), and
3. \(G_m = P_m; H_m = C_m\).

For a path \( P_n \) of order \( n \) and for any odd integer \( k \), \( 1 \leq k \leq n – 3 \), Chartrand et al. have given an upper bound for the radio \( k \)-chromatic number of \( P_n \) as \( \frac{k^2+2k+1}{2} \). Here we improve this bound for \( \frac{n-4}{2} \leq k < \frac{2n-5}{3} \) and \( \frac{2n-5}{3} \leq k \leq n-3 \). They are \( \frac{k^2+k+4}{2} \) and \( \frac{k^2+k+2}{2} \), respectively. Also, we improve the lower bound of Kchikech et al. from \( \frac{k^2+3}{2} \) to \( \frac{k^2+5}{2} \) for odd integer \( k \), \( 3 \leq k \leq n-3 \).