The first, second Zagreb connection indices and modified first Zagreb connection index are defined as \(Z{C_1}(G)={\sum\limits_{{v\in V(G)}} {{\tau _G}^2(v)} }\), \(ZC{}_{2}(G)=\displaystyle\sum_{uv\in E(G)}^{}\tau{}_{G}(u)\tau{}_{G}(v)\) and \(ZC{}_{1}^{\ast }(G)=\displaystyle\sum_{v\in V(G)}^{}d{}_{G}(v)\tau{}_{G}(v)\), respectively. In this paper, we consider the maximum values of \(Z{C_1}(G)\), \(Z{C_2}(G)\), \(Z{C_1}^{*}(G)\) of \(n\)-vertex trees with fixed matching number \(m\) and the extremal graphs are also characterized.

An improper interval (edge) coloring of a graph \(G\) is an assignment of integer colors to the edges of \(G\) satisfying the condition that, for every vertex \(v \in V(G)\), the set of colors assigned to the edges incident with \(v\) forms an integral interval. An interval coloring is \(k\)-improper if at most \(k\) edges with the same color all share a common endpoint. The minimum integer \(k\) such that there exists a \(k\)-improper interval coloring of the graph \(G\) is the interval coloring impropriety of \(G\), denoted by \(\mu_{int}(G)\). In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, \(G\odot H\).

A decomposition \(\mathcal{C}\) of a graph \(G\) is primitive if no proper, nontrivial subset of \(\mathcal{C}\) is a decomposition of an induced subgraph of \(G\). The existence of primitive decompositions has been studied for several decompositions, including path and cycle decompositions for complete and cocktail party graphs. In this work, we classify the existence of primitive star decompositions for complete graphs.

In this paper, we study the distribution on \([k]^n\) for the parameter recording the number of indices \(i \in [n-1]\) within a word \(w=w_1\cdots w_n\) such that \(|w_{i+1}-w_i|\ \leq 1\) and compute the corresponding (bivariate) generating function. A circular version of the problem wherein one considers whether or not \(|w_n-w_1|\ \leq 1\) as well is also treated. As special cases of our results, one obtains formulas involving staircase and Hertzsprung words in both the linear and circular cases. We make use of properties of special matrices in deriving our results, which may be expressed in terms of Chebyshev polynomials. A generating function formula is also found for the comparable statistic on finite set partitions with a fixed number of blocks represented sequentially.

Let \(G=(V,E)\) be a graph of order \(n\) without isolated vertices. A bijection \(f\colon V\rightarrow \{1,2,\dots,n\}\) is called a local distance antimagic labeling, if \(w(u)\not=w(v)\) for every edge \(uv\) of \(G\), where \(w(u)=\sum_{x\in N(u)}f(x)\). The local distance antimagic chromatic number \(\chi_{ld}(G)\) is defined to be the minimum number of colors taken over all colorings of \(G\) induced by local distance antimagic labelings of \(G\). The concept of Generalized Mycielskian graphs was introduced by Stiebitz [20]. In this paper, we study the local distance antimagic labeling of the Generalized Mycielskian graphs.

A graph is called \(t\)-existentially closed (\(t\)-e.c.) if it satisfies the following adjacency property: for every \(t\)-element subset \(A\) of the vertices, and for every subset \(B \subseteq A\), there exists a vertex \(x \in A\) that is adjacent to all vertices in \(B\) and to none of the vertices in \(A \setminus B\). A \(t\)-e.c. graph is critical if removing any single vertex results in a graph that is no longer \(t\)-e.c. This paper investigates \(2\)-e.c. critical Cayley graphs and vertex-transitive graphs, providing explicit constructions of \(2\)-e.c. critical Cayley graphs on cyclic groups. It is shown that a \(2\)-e.c. critical Cayley graph (as well as vertex-transitive graphs) of order \(n\) exists if and only if \(n \geq 9\) and \(n \notin \{10, 11, 14\}\). Additionally, this paper examines the numbers of \(2\)-e.c. (critical) vertex-transitive graphs among all vertex-transitive graphs for small orders, and presents detailed observations on some \(2\)-e.c. and \(3\)-e.c. vertex-transitive graphs.

For a family \(\mathcal F\) of graphs, a graph \(G\) is said to be \(\mathcal F\)-free if \(G\) contains no member of \(\mathcal F\) as an induced subgraph. We let \(\mathcal G_{3}(\mathcal F)\) be the family of \(3\)-connected \(\mathcal F\) -free graphs. Let \(P_{n}\) and \(C_{n}\) denote the path and the cycle of order \(n\), respectively. Let \(T_{0}\) be the tree of order nine obtained by joining a pendant edge to the central vertex of \(P_{7}\). Let \(T_{1}\) and \(T_{2}\) be the trees of order ten obtained from \(T_{0}\) by joining a new vertex to a vertex of \(P_{7}\) adjacent to an endvertex, and to a vertex of \(P_{7}\) adjacent to the central vertex, respectively. We show that \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{1}\})\) and \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{2}\})\) are finite families.

In this paper, we study additional aspects of the capacity distribution on the set ℬ_n of compositions of n consisting of 1’s and 2’s extending recent results of Hopkins and Tangboonduangjit. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on ℬ_n. We provide algebraic and combinatorial proofs of our results. We also give combinatorial explanations of some prior results where such a proof was requested. Finally, the joint distribution of the capacity statistic with two further parameters on ℬ_n is briefly considered.

We consider a ring \(R_{u^3} = \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2+u^3\mathbb{F}_2, u^4=0\). We discuss the structure of self-dual codes, Type I codes and Type II codes over the ring \(R_{u^3}\) in terms of the structure of their Torsion and Residue codes. We construct Type I and Type II optimal codes over the ring \(R_{u^3}\) for different lengths.

Magic squares are known to mankind since ages. With their eye catching properties, they have attracted and inspired researchers to further explore and work with them. The present paper is written with an aim to explore the usefulness of magic squares in the construction of partially balanced incomplete block (PBIB) designs. In this regard, we have proposed a new method for the construction of magic squares and studied their properties. We have also established a link between these properties and some existing association schemes. We have then introduced four new association schemes using these properties which have been later used for the construction of PBIB designs.