The matching energy of a graph was introduced by Gutman and Wagner in \(2012\) and defined as the sum of the absolute values of zeros of its matching polynomial. In this paper, we completely determine the graph with minimum matching energy in tricyclic graphs with given girth and without \(K_4\)-subdivision.

In this paper, we define and study the Gaussian Fibonacci and Gaussian Lucas \(p\)-numbers. We give generating functions, Binet formulas, explicit formulas, matrix representations, and sums of Gaussian Fibonacci \(p\)-numbers by matrix methods. For \(p = 1\), these Gaussian Fibonacci and Gaussian Lucas \(p\)-numbers reduce to the Gaussian Fibonacci and the Gaussian Lucas numbers.

Let \(G\) be a graph of order \(n\) and let \(Q(G, x) = \det(xI – Q(G)) = \sum_{i=0}^{n}(-1)^i\zeta_i(G)x^{n-i}\) be the characteristic polynomial of the signless Laplacian matrix of \(G\). We show that the Lollipop graph, \(L_{n,3}\), has the maximal \(Q\)-coefficients, among all unicyclic graphs of order \(n\) except \(C_n\). Moreover, we determine graphs with minimal \(Q\)-coefficients, among all unicyclic graphs of order \(n\).

Let \(G\) be a graph with \(n\) vertices, \(\mathcal{G}(G)\) the subdivision graph of \(G\). \(V(G)\) denotes the set of original vertices of \(G\). The generalized subdivision corona vertex graph of \(G\) and \(H_1, H_2, \ldots, H_n\) is the graph obtained from \(\mathcal{G}(G)\) and \(H_1, H_2, \ldots, H_n\) by joining the \(i\)th vertex of \(V(G)\) to every vertex of \(H_i\). In this paper, we determine the Laplacian (respectively, the signless Laplacian) characteristic polynomial of the generalized subdivision corona vertex graph. As an application, we construct infinitely many pairs of cospectral graphs.

In the paper, we show that the orientable genus of the generalized Petersen graph \(P(km, m)\) is at least \( \frac{km}{4} – \frac{m}{2}-\frac{km}{4m-4}+1\) if \(m\geq 4\) and \(k \geq 3\). We determine the orientable genera of \(P(3m, m)\), \(P(4k, 4)\), \(P(4m, m)\) if \(m \geq 4\), \(P(6m, m)\) if \(m \equiv 0 \pmod{2}\) and \(m \geq 6\), and so on.

Assume that \(\mu_1, \mu_2, \ldots, \mu_n\) are the eigenvalues of the Laplacian matrix of a graph \(G\). The Laplacian Estrada index of \(G\), denoted by \(LEE(G)\), is defined as \(LEE(G) = \sum_{i=1}^{n} e^{\mu_i}\). In this note, we give an upper bound on \(LEE(G)\) in terms of chromatic number and characterize the corresponding extremal graph.

In this note, we provide a combinatorial proof of a recent formula for the total number of peaks and valleys (either strict or weak) within the set of all compositions of a positive integer into a fixed number of parts.

The adjacent vertex distinguishing total chromatic number \(\chi_{at}(G)\) of a graph \(G\) is the smallest integer \(k\) for which \(G\) admits a proper \(k\)-total coloring such that no pair of adjacent vertices are incident to the same set of colors. Snarks are connected bridgeless cubic graphs with chromatic index \(4\). In this paper, we show that \(\chi_{at}(G) = 5\) for two infinite subfamilies of snarks, i.e., the Loupekhine snark and Blanusa snark of first and second kind. In addition, we give an adjacent vertex distinguishing total coloring using \(5\) colors for Watkins snark and Szekeres snark, respectively.

Let \(G\) be a tricyclic graph. Tricyclic graphs are connected graphs in which the number of edges equals the number of vertices plus two. In this paper, we determine graphs with the largest signless Laplacian spectral radius among all the tricyclic graphs with \(n\) vertices and diameter \(d\).