Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. An edge labeling $f:E\to Z_2$ induces a vertex labeling $f^{+}:V\to Z_2$ defined by $f^{+}(v)\equiv \sum _{uv\in E}f(uv)(\mathrm{mod}2)$, for each vertex $v\in V$. For $i\in Z_2$, let $v_f(i)=|\{v\in V:f^{+}(v)=i\}|$ and $e_f(i)=|\{e\in E:f(e)=i\}|$. An edge labeling $f$ of a graph $G$ is said to be edge-friendly if $|e_f(1)–e_f(0)|\le 1$. The set $\{v_f(1)–v_f(0):f\text{ is an edge-friendly labeling of }G\}$ is called the full edge-friendly index set of $G$. In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph $es(K_n)$ and a tight upper-bound for the complete $m$-ary tree ${es(T}_{m,h})$ using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular $k$-labeling of star $S_{m,n}$ and banana tree ${BT}_{m,n}$, new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for $\lceil \frac{|V|}{2}\rceil$-labeling.

A graph $G$ is called a fractional ID-$(g,f)$-factor-critical covered graph if for any independent set $I$ of $G$ and for every edge $e\in E(G-I)$, $G-I$ has a fractional $(g,f)$-factor $h$ such that $h(e)=1$. We give a sufficient condition using degree condition for a graph to be a fractional ID-$(g,f)$-factor-critical covered graph. Our main result is an extension of Zhou, Bian, and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-$k$-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013) 229–236] and Yashima’s previous result [T. Yashima, A degree condition for graphs to be fractional ID-$[a,b]$-factor-critical, Australasian Journal of Combinatorics 65(2016) 191–199].

In this article, we define $q$-generalized Fibonacci polynomials and $q$-generalized Lucas polynomials using $q$-binomial coefficient and obtain their recursive properties. In addition, we introduce generalized $q$-Fibonacci matrix and generalized $q$-Lucas matrix, then we derive their basic identities. We define $(k,q,t)$-symmetric generalized Fibonacci matrix and $(k,q,t)$-symmetric generalized Lucas matrix, then we give the Cholesky factorization of these matrices. Finally, we give determinantal and permanental representations of these new polynomial sequences.

Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley’s restriction is also considered.

For $r=1,2,\dots ,6$, we obtain generating functions $F_k^{(r)}(y)$ for words over the alphabet $[k]$, where $y$ tracks the number of parts and $[y^n]$ is the total number of distinct adjacent $r$-tuples in words with $n$ parts. In order to develop these generating functions for $1\le r\le 3$, we make use of intuitive decompositions but for larger values of $r$, we switch to the cluster analysis method for decorated texts that was introduced by Bassino et al. Finally, we account for the coefficients of these generating functions in terms of Stirling set numbers. This is done by putting forward the full triangle of coefficients for all the sub-cases where $r=5$ and 6. This latter is shown to depend on both periodicity and number of letters used in the $r$-tuples.

We consider the following variant of the round-robin scheduling problem: $2n$ people play a number of rounds in which two opposing teams of $n$ players are reassembled in each round. Each two players should play at least once in the same team, and each two players should play at least once in opposing teams. We provide an explicit formula for calculating the minimal numbers of rounds needed to satisfy both conditions. Moreover, we also show how one can construct the corresponding playing schedules.

Two binary structures $\mathfrak{R}$ and ${\mathfrak{R}}^{\prime }$ on the same vertex set $V$ are $(\le k)$-hypomorphic for a positive integer $k$ if, for every set $K$ of at most $k$ vertices, the two binary structures induced by $\mathfrak{R}$ and ${\mathfrak{R}}^{\prime }$ on $K$ are isomorphic. A binary structure $\mathfrak{R}$ is $(\le k)$-reconstructible if every binary
structure ${\mathfrak{R}}^{\prime }$ that is $(\le k)$-hypomorphic to $\mathfrak{R}$ is isomorphic to $\mathfrak{R}$. In this paper, we describe the pairs of $(\le 3)$-hypomorphic posets and the pairs of $(\le 3)$-hypomorphic bichains. As a consequence, we characterize the $(\le 3)$-reconstructible posets and the $(\le 3)$-reconstructible bichains. This answers a question suggested by Y. Boudabbous and C. Delhommé during a personal communication.

A tremendous amount of drug experiments revealed that there exists a strong inherent relation between the molecular structures of drugs and their biomedical and pharmacology characteristics. Due to the effectiveness for pharmaceutical and medical scientists of their ability to grasp the biological and chemical characteristics of new drugs, analysis of the bond incident degree (BID) indices is significant of testing the chemical and pharmacological characteristics of drug molecular structures that can make up the defects of chemical and medicine experiments and can provide the theoretical basis for the manufacturing of drugs in pharmaceutical engineering. Such tricks are widely welcomed in developing areas where enough money is lacked to afford sufficient equipment, relevant chemical reagents, and human resources which are required to investigate the performance and the side effects of existing new drugs. This work is devoted to establishing a general expression for calculating the bond incident degree (BID) indices of the line graphs of various well-known chemical structures in drugs, based on the drug molecular structure analysis and edge dividing technique, which is quite common in drug molecular graphs.

In this paper, we introduce a graph structure, called component intersection graph, on a finite dimensional vector space $\mathbb{V}$. The connectivity, diameter, maximal independent sets, clique number, chromatic number of component intersection graph have been studied.