Let \(G = (V(G), E(G))\) be a nonempty graph (may have parallel edges). The line graph \(L(G)\) of \(G\) is the graph with \(V(L(G)) = E(G)\), and in which two vertices \(e\) and \(e’\) are joined by an edge if and only if they have a common vertex in \(G\). We call the complement of \(L(G)\) as the jump graph. In this note, we give a simple sufficient and necessary condition for a jump graph to have a perfect matching.

We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not yield designs with new parameters, but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.

A shell graph of order \(n\), denoted by \(H(n, n-3)\), is the graph obtained from the cycle \(C_n\) of order \(n\) by adding \(n-3\) chords incident with a common vertex, say \(u\). Let \(v\) be a vertex adjacent to \(u\) in \(C_n\). Sethuraman and Selvaraju \([3]\) conjectured that for all \(k \geq 1\) and for all \(n_i \geq 4\), \(1 \leq i \leq k\), one edge \((uv)\) union of \(k\)-shell graphs \(H(n_i, n_i – 3)\) is cordial. In this paper, we settle this conjecture affirmatively.

In this paper, we give formulas for the sums of generalized order-\(k\) Fibonacci, Pell, and similar other sequences, which we obtain using matrix methods. As applications, we give explicit formulas for the Tribonacci and Tetranacci numbers.

A \((g, f)\)-coloring is a generalized edge-coloring in which each color appears at each vertex \(v\) at least \(g(v)\) and at most \(f(v)\) times, where \(g(v)\) and \(f(v)\) are nonnegative and positive integers assigned to each vertex \(v\), respectively. The minimum number of colors used by a \((g, f)\)-coloring of \(G\) is called the \((g, f)\)-chromatic index of \(G\). The maximum number of colors used by a \((g, f)\)-coloring of \(G\) is called the upper \((g, f)\)-chromatic index of \(G\). In this paper, we determine the \((g, f)\)-chromatic index and the upper \((g, f)\)-chromatic index in some cases.

The Szeged index extends the Wiener index for cyclic graphs by counting the number of atoms on both sides of each bond and summing these counts. This index was introduced by Ivan Gutman at the Attila Jozsef University in Szeged in \(1994\), and is thus called the Szeged index. In this paper, we introduce a novel method for enumerating by cuts. Using this method, an exact formula for the Szeged index of a zig-zag polyhex nanotube \(T = TUHC_6{[p,q]}\) is computed for the first time.

In this study, we showed that an \((n+1)\)-regular linear space, which is the complement of a linear space having points not on \(m+1\) lines such that no three are concurrent in a projective subplane of odd order \(m\), \(m \geq 9\), could be embedded into a projective plane of order \(n\) as the complement of Ostrom’s hyperbolic plane.

For general graphs \(G\), it is known \([6]\) that the minimal length of an addressing scheme, denoted by \(N(G)\), is less than or equal to \(|G| – 1\). In this paper, we prove that for almost all complete bipartite graphs \(K_{m,n}\), \(N(K_{m,n}) = |K_{m,n}| – 2\).

A vertex subversion strategy of a graph \(G\) is a set of vertices \(X \subseteq V(G)\) whose closed neighborhood is deleted from \(G\). The survival subgraph is denoted by \(G/X\). The vertex-neighbor-integrity of \(G\) is defined to be \(VNI(G) = \min\{|X| + r(G/X) : X \subseteq V(G)\},\) where \(r(G/X)\) is the order of a largest component in \(G/X\). This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. It was proved by Gambrell that the decision problem of computing the vertex-neighbor-integrity of a graph is NP-complete. In this paper, we evaluate the vertex-neighbor-integrity of the composition graph of two paths.

In this paper, we prove that a matroid with at least two elements is connected if and only if it can be obtained from a loop by a nonempty sequence of non-trivial single-element extensions and series extensions.