We show that if \(G\) has an odd graceful labeling \(f\) such that \(\max\{f(x): f(x) \text{ is even}, x \in A\} < \min\{f(x): f(x) \text{ is odd}, x \in B\}\), then \(G\) is an o-graph, and if \(G\) is an a-graph, then \(G \odot K_{n}\) is odd graceful for all \(w \geq 1\). Also, we show that if \(G_{1}\) is an a-graph and \(G_{2}\) is an odd graceful, then \(G_{1} \cup G_{2}\) is odd graceful. Finally, we show that some families of graphs are a-graphs and odd graceful.

Let \(K_{m} – H\) be the graph obtained from \(K_{m}\) by removing the edges set \(E(H)\) of \(H\) where \(H\) is a subgraph of \(K_{m}\). In this paper, we characterize the potentially \(K_{5} – P_{3}\), \(K_{5} – A_{3}\), \(K_{5} – K_{3}\) and \(K_{5} – K_{1,3}\)-graphic sequences where \(A_{3}\) is \(P_{2}\cup K_{2}\). Moreover, we also characterize the potentially \(K_{5} – 2K_{2}\)-graphic sequences where \(pK_2\) is the matching consisted of \(p\) edges.

Let \(G = (V, E)\) be a simple connected graph, where \(d_v\) is the degree of vertex \(v\). The zeroth-order Randić index of \(G\) is defined as \(R^0_n(G) = \sum_{v \in V} d_v^\alpha\), where \(\alpha\) is an arbitrary real number. Let \(G^*\) be the thorn graph of \(G\) by attaching \(d_G(v_i)\) new pendent edges to each vertex \(v_i\) (\(1 \leq i \leq n\)) of \(G\). In this paper, we investigate the zeroth-order general Randić index of a class thorn tree and determine the extremal zeroth-order general Randić index of the thorn graphs \(G^*(n,m)\).

Let \(X\) denote a set with \(q\) elements. Suppose \(\mathcal{L}(n, q)\) denotes the set \(X^n\) (resp. \(X^n \cup \{\Delta\}\)) whenever \(q = 2\) (resp. \(q \geq 3\)). For any two elements \(\alpha = (\alpha_1, \ldots, \alpha_n)\) and \(\beta = (\beta_1, \ldots, \beta_n) \in \mathcal{L}(n, q)\), define \(\alpha \leq \beta\) if and only if \(\beta = \Delta\) or \(\alpha_i = \beta_i\) whenever \(\alpha_i \neq 0\) for \(1 \leq i \leq n\). Then \(\mathcal{L}(n, q)\) is a lattice, denoted by \(\mathcal{L}_\bigcirc(n, q)\). Reversing the above partial order, we obtain the dual of \(\mathcal{L}_\bigcirc(n, q)\), denoted by \(\mathcal{L}_R(n, q)\). This paper discusses their geometricity, and computes their characteristic polynomials, determines their full automorphism groups. Moreover, we construct a family of quasi-strongly regular graphs from the lattice \(\mathcal{L}_\bigcirc(n, q)\).

A minimal separator of a graph is an inclusion-minimal set of vertices whose removal disconnects some pair of vertices. We introduce a new notion of minimal weak separator of a graph, whose removal merely increases the distance between some pair of vertices.

The minimal separators of a chordal graph \(G\) have been identified with the edges of the clique graph of \(G\) that are in some clique tree, while we show that the minimal weak separators can be identified with the edges that are in no clique tree. We also show that the minimal weak separators of a chordal graph \(G\) can be identified with pairs of minimal separators that have nonempty intersection without either containing the other—in other words, the minimal weak separators can be identified with the edges of the overlap graph of the minimal separators of \(G\).

A monitor is a computer in the network which is able to detect a fault computer among its neighbors. There are two stages of monitoring fault computer:(1) Sensing a fault among its neighbors and (2) Locating the fault computer.
A sensitive computer network requires double layer monitoring system where monitors are monitored. This problem is modeled using the graph theory concept of dominating set. In graph theory, there are two variations of domination concepts which represent double layer monitoring system.One concept is locating-domination and the other is liar domination.

It has been recently demonstrated that circulant network is a suitable topology for the design of On-Chip Multiprocessors and has several advantages over torus and hypercube from the perspectives of VLSI design. In this paper, we study both locating-domination and liar domination in circulant networks. In addition to characterization of locating-dominating set and liar dominating set of circulant networks, sharp lower and upper bounds of locating-dominating set and liar dominating set of circulant networks are presented.

We obtain some new examples of weakly distance-regular digraphs. Moreover, a class of commutative weakly distance-regular
digraphs of valency \(4\) and girth \(2\) is characterized.